= Solution
Let $\Lambda$ be a chiral gauge parameter. One consistent Abelian <supergauge transformation> convention is
$$
\Phi_i\mapsto e^{-iq_i\Lambda}\Phi_i,
\qquad
\bar\Phi_i\mapsto\bar\Phi_i e^{iq_i\bar\Lambda},
\qquad
V\mapsto V+i(\Lambda-\bar\Lambda),
$$
for which $\bar\Phi_i e^{q_iV}\Phi_i$ is invariant. <Wess-Zumino gauge> uses the nonordinary components of $\Lambda$ to remove the superfluous scalar and spinor components of $V$, leaving the photon, gauginos, complex scalar, and real auxiliary field of the two-dimensional vector multiplet. Ordinary $U(1)$ gauge transformations remain.
Define the field-strength multiplet, up to conventional normalization, by
$$
\boxed{\Sigma=\bar D_+D_-V}.
$$
Gauge invariance follows because chirality, antichirality, $\bar D_+^2=D_-^2=0$, and $\{\bar D_+,D_-\}=0$ annihilate the variation of $V$. The same identities give
$$
\bar D_+\Sigma=0,
\qquad
D_-\Sigma=0,
$$
so $\Sigma$ is a <twisted chiral superfield>. The <Fayet–Iliopoulos term> is a twisted F-term. In the standard axial convention its measure $d\theta^+d\bar\theta^-$ has axial charge $-2$, so invariance requires
$$
\boxed{q_A(\Sigma)=+2}.
$$
Reversing all axial-charge conventions reverses both signs but leaves this statement unchanged: the field and measure have opposite charges.
The charged matter fermions are chiral with respect to the axial symmetry. In a background with gauge flux, their <functional measure> has the two-dimensional axial anomaly
$$
\partial_\mu j_A^\mu=\frac1\pi\left(\sum_iq_i\right)F_{01}
$$
up to orientation and current normalization. Equivalently, a Fujikawa transformation multiplies the torus path integral by a phase proportional to $\alpha(\sum_iq_i)\int F$. Since the background flux may be nonzero, the continuous axial $U(1)$ survives quantum mechanically precisely when
$$
\boxed{\sum_iq_i=0}.
$$
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