Classification of kernel-M-solvable odd anti-holes

Determine precisely which odd anti-holes are kernel-$M$-solvable, meaning that every $M$-clique-acyclic orientation of the anti-hole admits a kernel.

Background

The paper disproves a prior claim that the seven-vertex anti-hole is kernel-$M$-solvable and proves that odd anti-holes of order at least nine are simple kernel-solvable but not kernel-solvable. The exact classification of odd anti-holes under the stronger kernel-$M$-solvability notion remains unresolved.