Baker–Rumely lower bound conjecture for the tau constant

Let $\Gamma$ be a metrized graph, let $\tau(\Gamma)$ denote its tau constant, and let $\ell(\Gamma)$ denote its total length.

Baker–Rumely's lower bound conjecture. There is a universal constant $C>0$ such that, for every metrized graph $\Gamma$,

$$\tau(\Gamma) \geq C\cdot \ell(\Gamma).$$

This conjecture asks for a uniform lower bound for the tau constant in terms of the total length. The paper's abstract states that the bound is proved for metrized graphs with edge connectivity at least $5$, while the general conjecture remains unresolved there.