Thomassen's coloring-growth conjecture for triangle-free planar graphs

Let $G$ be an $n$-vertex triangle-free planar graph. A proper 3-coloring of $G$ is a vertex coloring with three colors in which adjacent vertices receive different colors. Thomassen's coloring-growth conjecture. There exists a constant $c>1$ such that every $n$-vertex triangle-free planar graph has at least

$$c^{n^{\log_{9/2}3}}$$

distinct proper 3-colorings. Thomassen's original exponential lower-bound conjecture was disproved by examples with only $2^{15n/\log_2 n}$ 3-colorings; the stated power-law bound is conjectured to be tight because the paper's construction attains the exponent $\log_{9/2}3$ for infinitely many values of $n$.