Polynomial-time minor containment conjecture for fixed codes

Fix a code ${\mathcal M}$. Given a code ${\mathcal C}$ of length $n$, polynomial-time minor-containment conjecture. It is decidable in time polynomial in $n$ whether ${\mathcal C}$ contains ${\mathcal M}$ as a minor.

This is an algorithmic counterpart to finite excluded-minor characterizations: testing containment of a fixed minor would provide an efficient recognition procedure for the corresponding excluded-minor class. The source gives no resolution status.