DK conjecture for simple flips of homogeneous type

Let $X_{\alpha}$ and $X_{\beta}$ be the varieties associated with the simple flip in Proposition, and let $D(X_{\alpha})$ and $D(X_{\beta})$ denote their triangulated derived categories. DK conjecture for simple flips of homogeneous type. In the circumstance of Proposition, there exists a fully faithful functor

$$\Phi:D(X_{\beta})\hookrightarrow D(X_{\alpha}).$$

This is the homogeneous-type instance of the derived-category expectation for flips discussed in the paper. The supplied text does not establish whether this assertion is proved in full generality, so its status remains open.