Compatibility of global and critical obstruction theories for local Quot schemes

Let $X$ be the smooth toric threefold, let $U_\alpha\subset X$ be the affine chart associated with a torus fixed point, and let $F$ be the sheaf under consideration. Let ${\mathbb E}$ be the global perfect obstruction theory on $\operatorname{Quot}_X(F,n)$, and let

$$\iota_{n,\alpha}\colon \operatorname{Quot}_{U_\alpha}(F|_{U_\alpha},n)\hookrightarrow \operatorname{Quot}_X(F,n)$$

be the open immersion. Denote by ${\mathbb E}_{\operatorname{crit}}$ the critical obstruction theory on the local Quot scheme.

Compatibility conjecture. The restriction $\iota_{n,\alpha}^*{\mathbb E}$ agrees, as a symmetric perfect obstruction theory, with ${\mathbb E}_{\operatorname{crit}}$.

This would identify the global obstruction theory with the critical-locus obstruction theory on each local chart. The source notes that the general result is open, although the corresponding K-theory class comparison is known.