Self-avoiding-walk inequality beyond cut-edges

Determine whether the inequality $|\mathscr{S}(u_0,v_0)|\leq|\mathscr{S}(u_0,v_1)|$ holds for every finite, connected, simple graph $G$ and every pair of distinct vertices $u,v\in V(G)$ such that $(u,v)$ is not a cut-edge of $G$.

Background

The general self-avoiding-walk comparison posed earlier in the paper fails for ladder graphs and for pairs joined by suitable cut-edges. The authors’ decomposition shows that cut-edges can force strictly more walks from $u_0$ to $v_0$ than from $u_0$ to $v_1$.

The paper explicitly states that it is unknown whether cut-edges are the only obstruction. It therefore formulates the restricted question of whether the opposite-layer inequality always holds when the pair $(u,v)$ is not a cut-edge of the base graph.