The semigroup representation conjecture for generalized Petersen graphs

Let $G(n,k)$ be a generalized Petersen graph. A semigroup graph is a graph admitting a Cayley representation by a semigroup. The semigroup representation conjecture. Every generalized Petersen graph is a semigroup graph.

This asks whether the entire generalized Petersen family admits semigroup Cayley representations. The source states that the question remains open, while noting that the first cases $G(7,2)$ and $G(7,3)$ require a connection set of size $3$ in the monoid setting.