Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $K:=\overline{\mathbb{F}_p}$.

Let $b$ be a $p$-block of $G$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer correspondent of $b$.

M. Broué conjectured in the 90's that $b$ and $c$ are derived equivalent under these assumptions.

J. Rickard strengthened Broué's conjecture by conjecturing that equivalence can be given by a bounded chain complex of $p$-permutation bimodules whose indecomposable direct summands have vertices contained in $\Delta (D)$.

This means the following.

Let $Y$ be a virtual $KGb-KHc$-bimodule that is a linear combination of $p$-permutation bimodules that are finitely generated projective as left and right modules, satisfying $Y\otimes_{KHc}Y^{*}=[KGb]$ and $Y^{*}\otimes_{KGb}Y=[KHc]$ in the appropriate Grothendieck groups of bimodules.

More specifically, $Y\in T^{\Delta}(b,c)$, where $Y\in T^{\Delta}(b,c)$ denotes (in our context of Rickard's strengthening of Broué's abelian defect group conjecture) the Grothendieck group, w.r.t. $\oplus$, of the category of $p$-permutation $(b, c)$-bimodules, all of whose indecomposable direct summands have vertices contained in $\Delta(D) := \{(x, x) | x\in D\}$.

In particular, $Y$ induces a splendid virtual Morita equivalence.

Question:What is known about possible ways / strategies to (maybe computationally) lift such a splendid virtual Morita equivalence up to a splendid derived equivalence?If not in general, is there something known, if $Y$ has very few indecomposable direct summands?

I've seen in the literature so far that there is a related article by D. Craven and R. Rouquier (see https://doi.org/10.1016/j.aim.2013.07.010) where they provide lifts of stable equivalences to perverse equivalences and they construct derived equivalences from these.

I would be particularly interested in the question of what references in the literature contain known ways or ideas on how to attack the problem to lift splendid virtual Morita equivalences up to splendid derived equivalences, but, of course, any other related literature would also be gratefully appreciated.