Non-cut-edge criterion for self-avoiding walks on bunkbed graphs

Establish whether, for a finite, connected, simple graph $G$ and distinct vertices $u,v\in V(G)$ with $(u,v)$ not a cut-edge of $G$, the self-avoiding-walk inequality $|\mathscr{S}(u_0,v_0)|\leq|\mathscr{S}(u_0,v_1)|$ holds on $G\times K_2$.

Background

This is the paper’s explicitly stated modified open question after identifying cut-edges as an obstruction to the unrestricted comparison. It asks whether excluding cut-edges is sufficient to guarantee at least as many self-avoiding walks to the opposite layer as to the same layer.

The question applies to arbitrary finite, connected, simple base graphs and all distinct vertex pairs not joined by a cut-edge.