= Solution
\b[$W_\alpha$ is a fermionic chiral spinor superfield.] Its chirality follows directly from the <Abelian field-strength chiral projection>:
$$
\bar D_{\dot\beta}W_\alpha=-\frac14\bar D_{\dot\beta}\bar D^2D_\alpha V=0.
$$
There are only two independent barred <supersymmetric covariant derivatives>, and their equal-chirality <anticommutators> vanish. Every product of three barred derivatives therefore vanishes. Equivalently, the previous calculation has no independent barred <Grassmann variable> at fixed $y$. The free undotted index makes this a <chiral spinor superfield>, rather than a scalar <chiral superfield>; its lowest component is the odd <gaugino> $\lambda_\alpha$.
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