= Solution
Use the printed <hypercharge> convention; reversing every hypercharge relative to another common convention does not change the gauge invariants. Write $U,D,E,N$ for the singlet <chiral superfields> labelled with a superscript $R$ in the paper, and define the weak-doublet contraction $A\cdot B=\epsilon_{ab}A^aB^b$. The most general homogeneous <cubic superpotential with right-handed-neutrino superfields> separates as $W_3=W_{B,L}+W_{\rm viol}$, where
$$
\boxed{W_{B,L}=(y_u)_{ij}(Q_i\cdot H_2)U_j+(y_d)_{ij}(Q_i\cdot H_1)D_j
+(y_e)_{ij}(L_i\cdot H_1)E_j+(y_\nu)_{ij}(L_i\cdot H_2)N_j,}
$$
and
$$
\boxed{\begin{aligned}
W_{\rm viol}={}&\frac12\lambda_{ijk}(L_i\cdot L_j)E_k+\lambda'_{ijk}(L_i\cdot Q_j)D_k\\
&+\frac12\lambda''_{ijk}\epsilon^{abc}U_{ia}D_{jb}D_{kc}
+\eta_iN_i(H_1\cdot H_2)+\frac16\kappa_{ijk}N_iN_jN_k.
\end{aligned}}
$$
Repeated generation and color indices are summed. Because the <chiral superfields> commute while the gauge contractions are antisymmetric,
$$
\lambda_{ijk}=-\lambda_{jik},\qquad \lambda''_{ijk}=-\lambda''_{ikj},\qquad \kappa_{ijk}=\kappa_{(ijk)}.
$$
The factors $1/2$ and $1/6$ remove the corresponding overcounting.
Here is a completeness check, not merely a list of possible terms. A color-singlet cubic can have no colored fields, a fundamental–antifundamental pair, or three fundamentals or three antifundamentals. A colored pair must be $QU$ or $QD$: the remaining weak doublet and <hypercharge> select respectively $H_2$ or $H_1,L$. Three antifundamentals allow only $UDD$ at zero <hypercharge>; three $Q$ doublets have no weak singlet. With no colored fields, weak invariance requires zero or two doublets. Two positive-hypercharge doublets require $E$, giving $LH_1E$ and $LLE$, while $H_1\cdot H_1=0$. Opposite-hypercharge doublets require $N$, giving $LH_2N$ and $H_1H_2N$. With no doublets, only $NNN$ has zero <hypercharge>. These are exactly the nine structures above.
For <baryon number> and <lepton number>, assign
$$
B(Q)=\tfrac13,\quad B(U)=B(D)=-\tfrac13,\qquad L(L)=1,\quad L(E)=L(N)=-1,
$$
and zero charges to the Higgs fields. This is the conjugate-singlet convention, so the neutrino <Yukawa coupling> conserves <lepton number>. The first four terms conserve both charges. The $LLE$ and $LQD$ terms carry $L=1$, the $UDD$ term carries $B=-1$, the $NH_1H_2$ term carries $L=-1$, and $NNN$ carries $L=-3$. The gauge-singlet neutrino interactions must therefore be included in the violating sector.
If “cubic” instead means a polynomial of degree at most three, there are also the lower-degree gauge invariants
$$
W_{\le2}=W_0+t_iN_i+\frac12M_{ij}N_iN_j+\mu H_1\cdot H_2+\mu_iL_i\cdot H_2,\qquad M_{ij}=M_{ji}.
$$
The Higgs mass term conserves both charges; $t_iN_i$, $M_{ij}N_iN_j$ and $\mu_iL_i\cdot H_2$ violate <lepton number>. These additional terms distinguish the full renormalizable <superpotential> from its homogeneous cubic part.
To obtain <squark-mediated proton decay>, combine a <lepton number>-violating $LQD$ vertex with a <baryon number>-violating $UDD$ vertex. A nonzero example is $\lambda'_{112}$ with $\lambda''_{112}$, exchanging the singlet <strange quark> <squark> of mass $M_{\widetilde s}$. The antisymmetry forbids $\lambda''_{111}$, so using three first-generation singlets would not be a valid example. The relevant component vertices contain $e_Lu_L\widetilde s^{\,c}$ and $u^cd^c\widetilde s^{\,c}$, with the required color contraction. Connecting one vertex to the complex conjugate of the other through the massive scalar propagator produces schematically
$$
\mathcal L_{\rm eff}\supset\frac{\lambda'_{112}\lambda''^{*}_{112}}{M_{\widetilde s}^2}
\epsilon_{abc}(u_R^a d_R^b)(e_Lu_L^c)+\mathrm{h.c.}
$$
Parentheses denote Lorentz scalar two-spinor contractions, dotted for the right-handed pair and undotted for the left-handed pair. This dimension-six <proton-decay operator> contains the three quarks $uud$ and an electron field that can create a <Positron>. Its hadronic matrix element $\langle\pi^0|\epsilon_{abc}(u_R^ad_R^b)u_L^c|p\rangle$ gives the channel $p\to e^+\pi^0$. The internal strange flavor need not appear in the final state. The two interactions together violate both continuous charges, as required for this channel.
The coefficient of the dimension-six <effective operator> has mass dimension $-2$. Thus <dimensional analysis> gives
$$
\boxed{\Gamma(p\to e^+\pi^0)\sim c_h\,|\lambda'\lambda''|^2\frac{m_p^5}{M_{\widetilde s}^4},\qquad
\tau_p\sim\frac{M_{\widetilde s}^4}{c_h|\lambda'\lambda''|^2m_p^5},}
$$
where $c_h$ is a dimensionless hadronic and phase-space factor. Its detailed value cannot be fixed by dimensional reasoning. Applying the historical lower bound supplied in the paper gives the <proton-lifetime bound on a product of R-parity-violating couplings>
$$
\boxed{|\lambda'\lambda''|\lesssim\frac{M_{\widetilde s}^2}{\sqrt{c_h\,(2.4\times10^{64}\,\mathrm{GeV}^{-1})\,m_p^5}}
\simeq\frac{6.5\times10^{-27}}{\sqrt{c_h}}\left(\frac{M_{\widetilde s}}{1\,\mathrm{TeV}}\right)^2\left(\frac{1\,\mathrm{GeV}}{m_p}\right)^{5/2}.}
$$
The numerical coefficient is only the accuracy of the requested dimensional estimate; the squark mass was not fixed by the question.
Finally, <R-parity> is $R_p=(-1)^{3(B-L)+2s}$. At the superfield level use <matter parity> $P_M=(-1)^{3(B-L)}$: $Q,U,D,L,E,N$ are odd and $H_1,H_2$ are even. Each conserving <Yukawa coupling> contains two matter <chiral superfields> and one Higgs field, so it is even. Every violating homogeneous cubic above is odd: $LLE,LQD,UDD,NNN$ contain three matter fields, while $NH_1H_2$ contains one. \b[R-parity forbids all the listed cubic violations and preserves the Yukawa terms that generate fermion masses.] The Higgs bilinear is also even, whereas the lepton bilinear $LH_2$ and the singlet linear term are odd.
The final assertion must not be read as forbidding every possible violation of continuous <baryon number> or <lepton number>. In particular, <matter parity allows Majorana neutrino masses>: $N_iN_j$ is even but carries $L=-2$, a concrete counterexample once quadratic terms are included. Higher-dimensional operators such as $QQQL$ and $(LH_2)^2$ are likewise even and violate the continuous charges. The verified statement here concerns the homogeneous cubic sector, including the neutrino singlet terms.
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