Esperet's polylogarithmic induced-path conjecture for k-degenerate graphs

Let $k$ be a positive integer. A graph is $k$-degenerate if every subgraph has a vertex of degree at most $k$. For a graph with a path of order $n$, consider the order of its longest induced path.

Esperet's conjecture. There is a constant $d$ such that every $k$-degenerate graph that has a path of order $n$ also has an induced path of order at least

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

This conjecture asks whether the polylogarithmic lower bound is the correct general bound for induced paths in $k$-degenerate graphs. It is posed as an open problem in the source; the paper proves such a bound for broader structural classes, but not for all $k$-degenerate graphs.