Strict 1/108 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.

Strict $1/108$ lower bound conjecture. For every metrized graph $\Gamma$,

$$\tau(\Gamma)>\frac{\ell(\Gamma)}{108}.$$

The conjecture refines the Baker–Rumely lower bound by proposing the explicit sharp constant $1/108$. The source reports computational and theoretical evidence that tau constants can approach, but not attain, this value; no resolution is supplied in the paper.