Le's polynomial bound on induced paths in $\mathcal{O}_2$-free graphs

Let $n$ be a positive integer and let an $\mathcal{O}_2$-free graph be a graph containing no induced subgraph from the family $\mathcal{O}_2$. An induced path is a path whose vertices induce exactly the edges of the path.

Le's induced-path conjecture. There is a constant $c$ such that every $\mathcal{O}_2$-free $n$-vertex graph has at most $n^c$ distinct induced paths.

This conjecture was raised earlier as an open question by Raymond. The paper places it in the context of recognizing $\mathcal{O}_k$-free graphs and does not report a resolution.