Kuznetsov–Schinder's derived equivalence implies L-equivalence conjecture

Let $X$ and $Y$ be smooth projective simply connected varieties. They are L-equivalent if $[X]-[Y]$ vanishes in the localization $K_0(\operatorname{Var}_{\mathbb{C}})[\mathbb{L}^{-1}]$, equivalently if $([X]-[Y])\cdot\mathbb{L}^n=0$ for some $n>0$. Kuznetsov–Schinder's conjecture. If

$$D^b(X)\cong D^b(Y),$$

then $X$ and $Y$ are L-equivalent. The paper states that this is the original version of Kuznetsov–Schinder's conjecture and that it is disproved there; the published version was modified.