Coprimality of subset sizes attaining the n−1 extension bound

Prove that, for every synchronizing Eulerian binary automaton on n states, any proper nonempty subset S satisfying minext(S)=n−1 has cardinality coprime to n.

Background

Theorem 5.2 constructs, for every n≥4 and every subset size m coprime to n, a synchronizing Eulerian binary automaton possessing a subset of size m whose shortest extending word has length n−1. Exhaustive computations indicate that subset sizes not coprime to n attain at most n−2 on the Eulerian binary fibres examined.

The unresolved converse concerns composite n and binary automata with sufficiently many collisions. The paper proves a partial restriction: every prime divisor of gcd(|S|,n) is at most the maximum collision number of a letter. The conjecture is specifically binary, since the paper gives a ternary four-state counterexample.