Estrada–Mugnolo's Kirchhoff-index ordering conjecture for hubs-biased resistance distances

Let $G$ be a connected graph. Write $\mathcal{\mathcal{\mathscr{R}}_{\mathnormal{\alpha=1}}}(G)$ for the Kirchhoff index based on the hubs-repelling resistance distance, $\mathcal{\mathcal{\mathscr{R}}}_{\alpha=0}(G)$ for the standard Kirchhoff index, and $\mathcal{\mathcal{\mathscr{R}}_{\mathnormal{\alpha=-1}}}(G)$ for the Kirchhoff index based on the hubs-attracting resistance distance.

Kirchhoff-index ordering conjecture. For $\alpha\in\{-1,1\}$,

$$\mathcal{\mathcal{\mathscr{R}}_{\mathnormal{\alpha=1}}}(G)\geq\mathcal{\mathcal{\mathscr{R}}}_{\alpha=0}(G)\geq\mathcal{\mathcal{\mathscr{R}}_{\mathnormal{\alpha=-1}}}(G),$$

with equality if and only if $G$ is regular.

The conjecture was formulated after computational experiments on more than 12,000 connected graphs with $5\leq n\leq 8$, and predicts that hubs-repelling navigation gives the largest Kirchhoff index while hubs-attracting navigation gives the smallest. Its general validity is left open in the source.