The quotient-stack blowup F-bundle isomorphism conjecture

Let $X$ be a smooth projective variety with an action of a group $G$, and let $Z\subset X$ be a $G$-invariant submanifold of codimension $c\geq 2$. Write

$$\widetilde X=\operatorname{Bl}_Z(X),\qquad X'=X\sqcup Z\sqcup\cdots\sqcup Z$$

with $c-1$ copies of $Z$, and let $B_{[\widetilde X/G]}$ and $B_{[X'/G]}$ be the corresponding bases of F-bundles with their $\mathsf{MT}$-actions. Quotient-stack blowup conjecture. There is a canonical partial isomorphism $iso_{[\widetilde X/G],[X'/G]}$ of F-bundles, restricted to nonempty connected analytic domains in $B_{[\widetilde X/G]}$ and $B_{[X'/G]}$, that commutes with the $\mathsf{MT}$-action. This is proposed as an analogue for quotient stacks of Iritani's blowup theorem; the source expects it to follow from a sufficiently detailed understanding of the mechanism of that theorem.