Local girth correspondence-colourability conjecture for planar graphs

Let $G$ be a planar graph, and let $(L,M)$ be a correspondence assignment for $G$. The list assignment $L$ is a local girth list assignment when the list size at each vertex is determined by the girth of the relevant graph structure, as in the source.

Local girth correspondence-colourability conjecture. If $G$ is a planar graph and $(L,M)$ is a correspondence assignment for $G$ where $L$ is a local girth list assignment, then $G$ is $(L,M)$-colourable.

It is not currently known whether planar graphs are local girth correspondence colourable at all; this conjecture would establish that property.