The self-avoiding-walk bunkbed question

Suppose $G=(V,E)$ is a finite, connected, simple graph. For $x,y\in V$, let $\mathscr{S}(x,y)$ be the set of self-avoiding walks from $x$ to $y$, and let $\mathscr{S}(u_i,v_j)$ denote the corresponding set on $G\times K_2$. Self-avoiding-walk bunkbed question. Is it true that for all distinct $u,v\in V(G)$,

$$\big|\mathscr{S}(u_0,v_0)\big|\leq \big|\mathscr{S}(u_0,v_1)\big|?$$

The question has a negative answer in general: the inequality fails for ladder graphs, although it holds for complete bunkbed graphs $K_n\times K_2$ for all $n\geq 3$ by the stated theorem. Therefore the proposed assertion is refuted.