L-equivalence implies derived equivalence for projective hyperkähler manifolds

Let $X$ and $Y$ be projective hyperkähler manifolds. They are $L$-equivalent when $[X]-[Y]$ is annihilated by a power of $\mathbb{L}$ in $K_0(\operatorname{Var}_k)$, equivalently when $\mathbb{L}^r([X]-[Y])=0$ for some positive integer $r$.

L-equivalence conjecture for hyperkähler manifolds. If $X$ and $Y$ are $L$-equivalent, then their bounded derived categories of coherent sheaves are equivalent:

$$X\sim_LY\implies \mathcal{D}^{b}(X)\simeq\mathcal{D}^{b}(Y).$$

This is proposed as a partial converse to the Kuznetsov–Shinder conjecture. The source presents it as an open problem, and no resolution is supplied in the given text.