= Solution
Use left-chiral <superfields> throughout, abbreviating the conjugate singlets by $U_i=\bar u_i^c$, $D_i=\bar d_i^c$, $E_i=\bar e_i^c$ and $N_i=\bar\nu_i^c$. The printed <hypercharge> convention is the negative of the usual one, so $Q_{\mathrm{em}}=T_3-Y$; consequently $H_1$ is the down-type and $H_2$ the up-type Higgs doublet. Write $A\cdot B=\epsilon_{\alpha\beta}A^\alpha B^\beta$ for the weak-doublet contraction. Assign <baryon number> $B(Q)=1/3$, $B(U)=B(D)=-1/3$, and <lepton number> $L(L)=1$, $L(E)=L(N)=-1$; both Higgs fields have $B=L=0$.
The <baryon number> and <lepton number> conserving homogeneous cubic <superpotential> is
$$
\boxed{W_{3,\mathrm{cons}}=(y_u)_{ij}(Q_i\cdot H_2)U_j+(y_d)_{ij}(Q_i\cdot H_1)D_j+(y_e)_{ij}(L_i\cdot H_1)E_j+(y_\nu)_{ij}(L_i\cdot H_2)N_j.}
$$
Each term has zero <hypercharge>, is a color and weak singlet, and has $B=L=0$. Higgs expectation values give the usual <quark> and charged-lepton <Yukawa interaction> masses and, here, Dirac <neutrino> masses. The complete remaining cubic part is
$$
\begin{aligned}
W_{3,\mathrm{viol}}={}&\frac12\lambda_{ijk}(L_i\cdot L_j)E_k+\lambda'_{ijk}(L_i\cdot Q_j)D_k\\
&+\frac12\lambda''_{ijk}\epsilon^{abc}U_{i,a}D_{j,b}D_{k,c}
+\kappa_iN_i(H_1\cdot H_2)+\frac16\rho_{ijk}N_iN_jN_k.
\end{aligned}
$$
The first two terms have $\Delta L=+1$, the third has $\Delta B=-1$, the fourth has $\Delta L=-1$, and the last has $\Delta L=-3$. Their Hermitian conjugates reverse these signs. Commutativity of <chiral superfields> and the antisymmetric weak/color contractions give $\lambda_{ijk}=-\lambda_{jik}$ and $\lambda''_{ijk}=-\lambda''_{ikj}$; $\rho$ can be taken completely symmetric. All other coupling <tensors> are unrestricted complex <tensors>.
To check completeness, a cubic <color singlet> containing a <quark> doublet and an <antiquark> singlet must contain a colorless weak doublet; the allowed hypercharges give $QH_2U$, $QH_1D$ and $QLD$. Three color-antitriplet singlets can form a singlet with $\epsilon^{abc}$, but <hypercharge> permits only $UDD$. Three <quark> doublets cannot be a weak singlet. With no colored fields, an even number of weak doublets is necessary. Two doublets and a singlet give precisely $LH_1E$, $LLE$, $LH_2N$ and $H_1H_2N$; contractions of two identical Higgs <superfields> vanish. With no doublets the only neutral triple is $NNN$. Thus \b[$W_3=W_{3,\mathrm{cons}}+W_{3,\mathrm{viol}}$ is the most general homogeneous cubic <superpotential>] for the printed fields, including their singlet <neutrinos>.
If “cubic” is intended to mean a polynomial of degree at most three, also add
$$
W_{\leq2}=\mu H_1\cdot H_2+\epsilon_iL_i\cdot H_2+\frac12M_{ij}N_iN_j+t_iN_i+W_0,\qquad M_{ij}=M_{ji}.
$$
The $\mu$ term conserves both charges; the other nonconstant terms have respectively $\Delta L=1,-2,-1$. This distinction will matter for the printed claim about <R-parity>.
For <squark-mediated proton decay>, choose $\lambda'_{11k}$ and $\lambda''_{11k}$ with $k=2$ or $3$. The same down-type singlet <squark> can appear in both vertices; $k=1$ would give $\lambda''_{111}=0$ by antisymmetry. The first term includes an electron/up-quark/<squark> vertex, and the second an up-quark/down-quark/<squark> vertex. Exchanging a <squark> of mass $M$ and expanding its propagator at momenta much smaller than $M$ produces the dimension-six <effective operator>
$$
\mathcal L_{\mathrm{eff}}\supset C\,\epsilon_{abc}(u_L^a e_L)(u_R^b d_R^c)+\mathrm{h.c.},\qquad C\sim\frac{\lambda'_{11k}\lambda''_{11k}{}^*}{M^2}.
$$
The left- and right-handed pairs are separately contracted spinor bilinears. This operator converts an up/down <quark> pair into a positron and an <antiquark>; the spectator up <quark> can combine with the <antiquark> into the neutral <pion>. Equivalently its hadronic <matrix> element between a <proton> and a <pion> is permitted, so it induces $p\to e^+\pi^0$. Having both the <baryon number> and <lepton number> violating vertices is what enables this channel.
Since $C$ has mass dimension $-2$, <dimensional analysis> gives
$$
\boxed{\Gamma(p\to e^+\pi^0)\sim c_h\frac{|\lambda'_{11k}\lambda''_{11k}|^2}{M^4}m_p^5.}
$$
The dimensionless $c_h$ includes the hadronic <matrix> element and two-body phase space; the <pion> mass changes this factor rather than the leading dimensions. Dimensional grounds alone do not give a precise hadronic coefficient. Using the lifetime supplied in the paper, rather than substituting a different bound, gives $\Gamma<1/(2.4\times10^{64})\,\mathrm{GeV}=4.17\times10^{-65}\,\mathrm{GeV}$ and hence
$$
\boxed{|\lambda'_{11k}\lambda''_{11k}|<\frac{M^2}{\sqrt{c_h(2.4\times10^{64}\,\mathrm{GeV}^{-1})m_p^5}}\simeq6.5\times10^{-27}c_h^{-1/2}\left(\frac{M}{1\,\mathrm{TeV}}\right)^2\left(\frac{1\,\mathrm{GeV}}{m_p}\right)^{5/2}.}
$$
For an illustrative $100\,\mathrm{GeV}$ <squark> this is about $6.5\times10^{-29}c_h^{-1/2}$ when $m_p$ is rounded to $1\,\mathrm{GeV}$. The bound constrains the relevant flavour product and depends on the exchanged mass; separate smallness of each coupling does not follow without an additional assumption.
Define <matter parity> and <R-parity> by
$$
\boxed{P_M=(-1)^{3(B-L)},\qquad R_p=(-1)^{3(B-L)+2s}=P_M(-1)^{2s}.}
$$
Every matter <superfield> $Q,U,D,L,E,N$ has $P_M=-1$, whereas $H_1,H_2$ and the <vector superfields> have $P_M=+1$. For ordinary <quarks> and <antiquarks>, $3(B-L)=\pm1$ is odd and $2s=1$, so $R_p=+1$. For charged <leptons>, <neutrinos> and their antiparticles, $3(B-L)=\mp3$ is odd and $2s=1$, again giving $R_p=+1$; the singlet <neutrino> fermion obeys the same rule. <Gluons>, the <photon>, $W$ and $Z$ have $B=L=0$ and spin one, hence $R_p=+1$; <Higgs bosons> have $B=L=s=0$ and also $R_p=+1$. Thus \b[every <Standard Model> particle is R-even], including the corresponding antiparticles. Changing spin by one half within a <supermultiplet> reverses $R_p$, so <squarks>, <sleptons>, <gauginos> and <Higgsinos> are R-odd.
Each conserving cubic term contains two R-odd matter <superfields> and one even Higgs <superfield>, so it is even under <matter parity>. Each term in $W_{3,\mathrm{viol}}$ has an odd number of matter <superfields>: three for $LLE,LQD,UDD,NNN$, and one for $NH_1H_2$. All are therefore forbidden. At the component level the two fermion spin factors in a <Yukawa interaction> multiply to $+1$, so the same selection rule follows directly from <R-parity>. This verifies \b[all cubic baryon/lepton-number violating terms are forbidden while the Yukawa fermion mass terms survive].
The blanket assertion needs a qualification beyond homogeneous cubic terms. The Higgs $\mu$ term is even; $L H_2$ and $N$ are odd and forbidden. But $N_iN_j$ is even and permits a Majorana <neutrino> mass despite violating <lepton number> by two. Likewise the higher-dimensional operators $QQQL$ and $(LH_2)^2$ are even. Consequently <R-parity> alone does not enforce all-order conservation of <baryon number> and <lepton number>, or absolute <proton> stability; the stated precise exclusion holds for the violating cubic interactions above.
Two physical implications follow from multiplicative conservation of <R-parity>. \b[<Superpartners> are produced in even numbers from an initial state consisting only of <Standard Model> particles], leading to pair production at the lowest threshold and parity-conserving cascades. \b[The <lightest supersymmetric particle> is stable], since one R-odd particle cannot decay exclusively into R-even particles and no lighter R-odd state is available. A neutral weakly interacting <lightest supersymmetric particle> can therefore be a dark-matter candidate and produce missing-energy signatures; neutrality and a suitable abundance require further model assumptions.
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