Gaiotto's nonvanishing integral conjecture

Let $W\in S^d\mathbb C^{n*}$ be a homogeneous polynomial, and define the real number

$$I(W):=\int_{\mathbb C^n}{\rm e}^{-|z|^2+W(z)-\overline{W(z)}}{\rm d}z{\rm d}\overline z.$$

Gaiotto's conjecture. The number $I(W)$ is strictly positive for all $W$, or equivalently, nonzero for all $W$. This is the $N=0$ case of part (ii) of the preceding conjecture, so it is a special case of the same proposed filtered-nondegeneracy statement rather than an independent claim. The supplied text gives no resolution of this special case.