Palette-sparsification conjecture at the local degree threshold

Let $G$ be a graph on vertex set $V$ with degree $d_v$ at each vertex $v$. Let $\Gamma$ be an arbitrary color set, let $S_v\subseteq\Gamma$ satisfy $|S_v|=d_v+1$ for every $v\in V$, and let each $L_v$ be a uniformly random $\ell$-subset of $S_v$, where

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

for fixed $\varepsilon>0$.

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

The source describes this as a further suspected extension that removes the global maximum-degree palette size and uses the natural local threshold. No resolution is given, so it remains open.