The avoidable induced path conjecture

Let $G$ be a graph, and let $P_k$ denote an induced path on $k$ vertices. An induced path $P$ is avoidable if every extension of $P$ is contained in an induced cycle, where an extension is an induced path obtained by adding one edge to $P$ at each end. Avoidable induced path conjecture. For every $k\ge 1$, every graph that contains an induced $P_k$ also contains an avoidable induced $P_k$. This conjecture would simultaneously generalize the avoidable-vertex theorem and the result that graphs with no induced cycles of lengths at least $k+3$ contain a simplicial induced $P_k$; its status is not resolved in the supplied source.