Wendland's five-list conjecture for facial unique-maximum coloring

Let $G$ be a plane graph. A FUM-coloring is a vertex coloring in which, on every face, the maximal color appearing on that face occurs at exactly one vertex. Suppose each vertex $v$ is assigned a list $L(v)$ of five integers.

Wendland's conjecture. If each vertex of a plane graph is assigned a list of $5$ integers, then there exists a FUM-coloring assigning each vertex a color from its list.

Wendland proved that lists of size $7$ suffice for FUM-coloring of every plane graph. Whether lists of size $5$ always suffice remains open.