The two-edge rooted-path inherence conjecture

A two-rooted graph $(H,s,t)$ is a graph $H$ with two distinguished vertices $s$ and $t$. A two-rooted graph is inherent if every graph containing a copy of $H$ also contains an avoidable copy of $(H,s,t)$. Let

$$P=(s,t,v)$$

be a two-edge path.

Two-edge rooted-path conjecture. The two-rooted path $(P,s,t)$ is inherent.

This is presented as the first nontrivial step toward understanding inherent graphs, following results that characterize necessary conditions for connected inherent two-rooted graphs and establish that all endpoints-rooted paths are inherent. Its resolution is not given in the source.