Exponential DP-4-colorings of triangle-free planar graphs

Let $G$ be a triangle-free planar graph with $n$ vertices, and let $P_{DP}(G,4)$ denote its DP color function. The exponential DP-coloring conjecture. There exists a constant $c>1$ such that, for every triangle-free planar $n$-vertex graph $G$,

$$P_{DP}(G,4)\geq c^n.$$

Triangle-free planar graphs are always 4-choosable and DP-4-colorable, although some are not 3-choosable and hence not DP-3-colorable. The conjecture asks for an exponential lower bound on the number of DP-4-colorings; the source contrasts it with subexponential upper bounds known for the number of 3-colorings.