The component-deletion conjecture for inherent two-rooted graphs

A two-rooted graph $(H,s,t)$ is a graph $H$ with two distinguished vertices $s$ and $t$. It is inherent if every graph containing a copy of $H$ also contains an avoidable copy of $(H,s,t)$. For graphs $H$ and $H'$, write $H+H'$ for their disjoint union, and suppose that $s,t$ are vertices of $H$.

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

Together with the component-separation conjecture, this addresses which disconnected two-rooted graphs can be inherent. The source does not provide a resolution.