Converse to derived isogeny implying quasi-liftable isogeny for abelian surfaces

Let $X$ and $Y$ be abelian surfaces over $\bar{k}$ with finite height and $\operatorname{char}(k)\neq 2$. They are quasi-liftably isogenous if they are isogenous through suitable liftings to characteristic zero. Converse conjecture. If $X$ and $Y$ are quasi-liftable isogenous, then they are derived isogenous; equivalently, under these assumptions, $X$ and $Y$ are derived isogenous if and only if they are quasi-liftable isogenous.

The forward implication is established in the cited theorem, and the authors state that they believe the converse also holds. Thus the converse remains open.