Partial-tilting extension conjecture for toric vector bundles

Let $\Sigma$ be a complete, simplicial fan such that the toric Deligne–Mumford stack $\mathcal{X}_\Sigma$ has a partial tilting complex $\mathcal{T}$ with finite global dimension endomorphism algebra. Let

$$\pi:\mathcal{V}\rightarrow X_\Sigma$$

be a toric vector bundle with fan $\Sigma_\mathcal{V}$. Suppose

$$H^i(\mathcal{X}_\Sigma,\mathcal{T}^\vee\otimes\mathcal{T}\otimes\operatorname{Sym}^\bullet(\mathcal{E}^\vee))=0$$

for all $i\neq0$. Partial-tilting extension conjecture. There is a partial tilting complex $\mathcal{T}'$ on $\mathcal{X}_{\Sigma_\mathcal{V}}$ whose endomorphism algebra has finite global dimension. Consequently, the ring $R=k[|\mathcal{V}|^\vee\cap M]$ admits an NCCR. The conjecture generalizes the preceding tilting-complex result to partial tilting complexes; the source does not provide evidence of a resolution.