The monoid Cayley characterization conjecture for generalized Petersen graphs

Let $G(n,k)$ be a generalized Petersen graph, and let ${\rm Cay}(M,C)$ denote a monoid Cayley graph with connection set $C$ of size $2$. The monoid Cayley characterization conjecture. The graph $G(n,k)$ is a monoid graph ${\rm Cay}(M,C)$ with $|C|=2$ if and only if one of the following holds:

$$(n,k)=(5,2),$$ $$(n,k)=(10,3),$$ $$k^2\equiv 1 \pmod n,$$

or

$$k^2\equiv \pm k \pmod n.$$

This claim gives a proposed complete classification of generalized Petersen graphs admitting a monoid Cayley representation with a two-element connection set; the source provides no resolution evidence beyond presenting it as a conjecture.