Tight cover-time conjecture for minimum-degree weighting

Let $G_w$ be a graph equipped with the minimum-degree weighting scheme, let $n$ be its number of vertices, and let $d$ be the parameter used in the locally tree-like analysis. Let $\text{{\bf COV}}[G_w]$ denote the resulting weighted random walk's cover time. Tight minimum-degree weighted cover-time conjecture. The bound in the source can be replaced by

$$\text{{\bf COV}}[G_w] \leq (1+o(1))\frac{d-1}{d-2}\;n\log n.$$

This is presented as a sharper bound than the preceding weighted cover-time estimate. The source gives no resolution.