Partial-tilting pullback conjecture for quotient stacks

Let $X$, $G$, and $\mathcal{E}$ be as in the surrounding construction, with $G$ finite, and let $\pi:[\mathbb{A}(\mathcal{E})/G]\rightarrow[X/G]$ be the induced projection. Suppose $\mathcal{T}$ is a partial tilting complex on $[X/G]$ whose endomorphism algebra has finite global dimension. Assume

$$H^i(X,\mathcal{T}^\vee\otimes\mathcal{T}\otimes S^l(\mathcal{E}))=0$$

for all $i\neq0$ and all $l>0$. Partial-tilting pullback conjecture. The pullback $\pi^*\mathcal{T}$ is a partial tilting complex on $[\mathbb{A}(\mathcal{E})/G]$ whose endomorphism algebra has finite global dimension. The conjecture proposes a quotient-stack generalization of the preceding tilting result; the source does not state that it has been resolved.