The concise tree-counting conjecture for planar graphs

Let $G=(V,E)$ be a simple planar graph, and let $tr(G)$ be the number of trees in $G$ of all sizes. Planar tree-counting conjecture. The function $tr$ is concise: every positive integer should be realized by a planar graph whose size is bounded by a fixed polynomial in the logarithm of that integer. The function is already almost complete because an empty graph on $n$ vertices has $n$ trees, and its computation is $\#\mathrm P$-complete, but conciseness remains open.