Palette-sparsification conjecture for arbitrary palettes

Let $G$ be a graph on $n$ vertices with maximum degree $\Delta_G\leq D$. Let $\Gamma$ be an arbitrary color set, let $S_v\subseteq\Gamma$ satisfy $|S_v|=D+1$ for every vertex $v$, and let each $L_v$ be a uniformly random $\ell$-subset of $S_v$, where

$$\ell=(1+\varepsilon)\log n$$

for fixed $\varepsilon>0$.

Palette-sparsification conjecture. The graph $G$ is $L$-colorable with high probability.

This is presented as an expected extension of the paper's main theorem and of the Alon–Assadi framework. The source gives no resolution, so the conjecture remains open.