Conjecture that every region in the fixed-multiplicity algorithm is empty

Let $m\geq 3$ and consider the regions arising in Algorithm~ for testing numerical semigroups of multiplicity $m$. Empty-region conjecture. For each $m\geq 3$, every region considered in Algorithm~ is empty.

The conjecture would imply Wilf's conjecture for numerical semigroups. The algorithm has been verified computationally through $m\leq 18$, and the source notes that the regions tested for $m\leq 87$ are in fact empty; the general assertion remains open.