Weakening conjecture for compatible-inclusion problems

Let $c\geq 1$ be an odd integer, and let $(P,Q)$ be a compatible-inclusion problem, with its $c$-weakening defined by retaining directed edges represented by paths of length at most $c$. The compatible-inclusion weakening conjecture. There is an odd integer $c\geq 1$ such that, for any compatible-inclusion problem $\mathcal{P}$, if $\mathcal{P}$ has a solution, then the $c$-weakening of $\mathcal{P}$ has a simple solution.

This is a combinatorial reformulation related to the upset matching conjecture. The paper notes that the case $c=3$ weakens a conjecture from the author's PhD thesis, but the supplied text gives no resolution of the present conjecture.