The four-list conjecture for facial unique-maximum edge-coloring

Let $G$ be a plane graph. A FUM-edge-coloring is a facial edge-coloring in which every face is incident with a unique edge colored with the maximal color appearing on that face. Suppose each edge $e$ is assigned a list $L(e)$ of four integers.

Four-list conjecture. If each edge of a plane graph is assigned a list of $4$ integers, then there exists a FUM-edge-coloring assigning each edge a color from its list.

This is the list-coloring analogue of the proposed four-color bound for FUM-edge-coloring. The source presents it as an expectation and gives no resolution, so it remains open.