3 problems

Beisegel et al.'s conjecture. For every positive integer $k$, every graph that contains an induced $k$-vertex path also contains an avoidable induced $k$-vertex path.

Component-deletion conjecture. If $(H+H',s,t)$ is inherent and $s,t$ are in $H$, then $(H,s,t)$ is also inherent.

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