Cotangent-stack invariance conjecture for Coulomb and Higgs categories

Let $\calY$ and $\calY'$ be stacks such that their cotangent dg-stacks are isomorphic as symplectic dg-stacks. Write $\calC_C(\calY)$ and $\calC_H(\calY)$ for the corresponding Coulomb and Higgs factorization categories, and $\calF_C(\calY)$ for the canonical Coulomb object. Cotangent-stack invariance conjecture. The corresponding $\mathbb Z_2$-graded versions of $\calC_C(\calY)$ and $\calC_C(\calY')$ are equivalent as $\mathbb Z_2$-graded factorization categories, with an equivalence sending $\calF_C(\calY)$ to $\calF_C(\calY')$; the analogous statement holds for $\calC_H$. This suggests that these categorical structures depend, at least after forgetting the integer grading, only on the symplectic cotangent dg-stack; the source gives no resolution.