Langhede and Thomassen's exponential correspondence-colouring conjecture for planar graphs

Let $G$ be a planar graph. A correspondence assignment $(L,M)$ for $G$ consists of a list assignment $L$ and, for each edge, a partial matching between the corresponding vertex-colour pairs; an $(L,M)$-colouring chooses a colour from each list while avoiding matched pairs on every edge. The class of planar graphs has exponentially many $5$-correspondence colourings if there exists a constant $c>0$ such that every planar graph $G$ and every $5$-correspondence assignment $(L,M)$ admit at least $2^{c\cdot v(G)}$ distinct $(L,M)$-colourings.

Langhede and Thomassen's conjecture. Planar graphs have exponentially many $5$-correspondence colourings.

This conjecture asks for an exponential strengthening of the known $5$-correspondence-colourability of planar graphs. The source gives no resolution, so the conjecture remains open.