Perfect tunneling conjecture for asymmetric equilateral triple wells

Let $x,y,z\in X$ be three wells in a graph with equal pairwise distances $a=b=c$, and let $c_{uv}$ denote the sum over shortest paths from $u$ to $v$ of the products of inverse square roots of the degrees of consecutive vertices. Assume $c_{xy}=c_{xz}\neq c_{yz}$. Perfect tunneling conjecture. There is always perfect tunneling from $y$ to $z$. This conjecture concerns the remaining asymmetric case of the equilateral triple-well setup, after distinguishing whether the relevant quantity $\sqrt{c_{yz}^2+8c_{xz}^2}$ is rational or irrational; the stated theorem gives different tunneling-time behavior in those cases, but does not resolve this assertion.