Self-avoiding-walk inequality for complete bunkbed graphs

Prove that the inequality between self-avoiding-walk counts, $|\mathscr{S}(u_0,v_0)|\leq|\mathscr{S}(u_0,v_1)|$, holds for the complete bunkbed graph $K_n\times K_2$ for every integer $n\geq3$.

Background

The paper studies self-avoiding walks on bunkbed graphs $G\times K_2$, comparing walks from $u_0$ to the same-layer vertex $v_0$ with walks from $u_0$ to the opposite-layer vertex $v_1$. For the complete base graph $K_n$, the authors prove equality when $n=2$ and strict inequality in favor of the opposite-layer endpoint for $n\in\{3,4,5\}$ and for all sufficiently large $n$.

The conjecture asks whether this inequality holds uniformly for every remaining complete-graph size $n\geq3$, thereby extending the proven finite and asymptotic cases to all complete bunkbed graphs.