Pathwidth–treedepth conjecture for graphs without long paths

Let $G$ be a graph, let $p$ be its pathwidth, and let $\ell$ be a positive integer. A path of order $2^\ell$ is a path with $2^\ell$ vertices. Pathwidth–treedepth conjecture. Every graph with pathwidth $p$ that contains no path of order $2^\ell$ has treedepth $O(p\ell)$. This would sharpen the known bound for treedepth in terms of treewidth, the height of a forbidden complete-binary-tree subdivision, and the length of a forbidden path; the source presents it as an open conjecture and gives no resolution.