= Solution
Choose a positive-energy massless momentum $P^\mu=(E,0,0,E)$. Then $\sigma^\mu P_\mu=E(I-\sigma^3)$, so the <Super-Poincaré algebra> gives
$$
\{Q_1,Q_1^\dagger\}=0,\qquad\{Q_2,Q_2^\dagger\}=4E.
$$
Unitarity forces $Q_1$ and its adjoint to vanish on this representation: the sum of their squared state norms is zero. Set $a=Q_2/\sqrt{4E}$. The surviving operators obey
$$
\{a,a^\dagger\}=1,\qquad a^2=(a^\dagger)^2=0.
$$
The <helicity> operator is $h=J_3$ for this momentum. In the component convention above, $[h,a]=a/2$ and $[h,a^\dagger]=-a^\dagger/2$. A highest-helicity state $|\lambda\rangle$ with $a|\lambda\rangle=0$ therefore generates exactly two states,
$$
\boxed{\bigl(\lambda,\lambda-\tfrac12\bigr),}
$$
one bosonic and one fermionic. This is the $\mathcal N=1$ <helicity spectrum of a massless supermultiplet>. The <CPT theorem> reverses helicity and conjugates internal charges, so <CPT completion of a supermultiplet> adds $(-\lambda,-\lambda+1/2)$ when that pair is not already present.
For the usual integer and half-integer helicities up to spin two, the CPT-complete contents are
$$
\begin{array}{c|c|c}
\text{multiplet}&\text{bosonic helicities}&\text{fermionic helicities}\\\hline
\text{chiral}&0,0&+\tfrac12,-\tfrac12\\
\text{vector}&+1,-1&+\tfrac12,-\tfrac12\\
\text{gravitino}&+1,-1&+\tfrac32,-\tfrac32\\
\text{gravity}&+2,-2&+\tfrac32,-\tfrac32.
\end{array}
$$
The <chiral superfield> describes one complex scalar and one Weyl fermion; the <vector multiplet> has a massless gauge vector and its gaugino. The gravitino multiplet contains a spin-one vector and a spin-three-halves field, while the <supergravity multiplet> contains the graviton and gravitino. All have two physical bosonic and two physical fermionic helicity states after CPT completion. The general two-state construction also describes higher-helicity formal multiplets; their existence as representations alone does not establish a consistent interacting higher-spin theory.
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