Twisted-cycle obstruction to PMM-griddability

Let $M$ be a partial multiplication matrix, meaning a matrix whose nonzero entries are monotone classes, and suppose its cell graph is cyclic. Let $\mathcal X$ be a permutation class monotone griddable by $M$. Twisted-cycle conjecture. The class $\mathcal X$ is monotone griddable by a partial multiplication matrix if and only if the corresponding class of permutation graphs does not contain arbitrarily wide constructions of the twisted cycles in a chain shown in Figure~, or their complements. The preceding theorem proves the relevant implication in a restricted setting; the claimed characterization for cyclic cell graphs remains open.