Tilting-type equivalence for the remaining stratified Atiyah flops

Let $\operatorname{Gr}(r,N)$ be the Grassmannian appearing in the stratified Atiyah flop, and suppose that $2r=N-1$ or $2r=N$. Expected tilting-type equivalence. There exists a tilting-type equivalence for the stratified Atiyah flop on $\operatorname{Gr}(r,N)$. The result is proved in the source when $2r\leq N-2$; the displayed assertion concerns the remaining cases $2r=N-1$ and $2r=N$, which the author expects to hold but does not establish there.