Esperet–Lemoine–Maffray conjecture on induced paths in degenerate graphs

Let $k\in\mathbb{N}$, and let $G$ be a $k$-degenerate graph with a path of order $n\geq 1$. An induced path is a path whose vertices induce exactly the edges of the path in $G$. Esperet–Lemoine–Maffray's conjecture. For every $k\in\mathbb{N}$ there is a constant $c>0$ such that $G$ has an induced path of order at least

$$(\log n)^c.$$

This conjecture asks for a polylogarithmic lower bound on the order of an induced path in every bounded-degeneracy graph containing a path of order $n$. The known general lower bound is of order $\log\log n$, while planar graphs and more generally graphs of bounded Euler genus satisfy stronger polylogarithmic bounds; the conjecture remains open in the stated generality.