Gaiotto's positivity and filtered nondegeneracy conjecture

Let $V=\mathbb C[z_1,\dots,z_n]$ be filtered by polynomial degree, with $F_NV=\mathbb C[z_1,\dots,z_n]_{\le N}$. Let $W\in S^d\mathbb C^{n*}$ be a homogeneous polynomial of degree $d$, and define

$$(P,Q)_W=\int_{\mathbb C^n}P(z)\overline {Q(z)}{\rm e}^{-|z|^2+W(z)-\overline{W(z)}}{\rm d}z{\rm d}\overline z.$$

When this sesquilinear form is filtered-nondegenerate, let $\langle\,,\rangle_W$ denote the resulting homogeneous form on $\mathbb C[z_1,\dots,z_n]$. Gaiotto's conjecture. (i) If $(\,,\,)_W$ is filtered-nondegenerate, then $\langle\,,\rangle_W$ is positive definite. (ii) The form $(\,,\,)_W$ is filtered-nondegenerate for all $W\in S^d\mathbb C^{n*}$. The claim generalizes Gaiotto's proposal for quasi-homogeneous superpotentials and concerns positivity arising from Landau--Ginzburg boundary conditions for 3d free hypermultiplets. Filtered nondegeneracy is known for sufficiently generic $W$, but the assertion for every homogeneous $W$ is not established in the supplied text; part (i) follows from part (ii) by deformation, and even follows from filtered nondegeneracy on a dense connected subset.