Baker and Rumely's tau lower bound conjecture for metrized graphs

Let $\Gamma$ be a metrized graph, let $\ell(\Gamma)=\int_{\Gamma} dx$ be its total length, and consider metrized graphs with $\ell(\Gamma)\neq 0$. Baker and Rumely's tau lower bound conjecture.

$$\text{inf}_{\Gamma}\frac{\tau(\Gamma)}{\ell(\Gamma)}>0,$$

where the infimum is taken over all metrized graphs $\Gamma$ with nonzero total length. The conjecture asserts the existence of a universal positive lower bound for the tau constant normalized by total length. The paper states a refinement asserting the explicit bound $\tau(\Gamma)>\frac{1}{108}\ell(\Gamma)$ for every metrized graph; the abstract describes $\frac{1}{108}$ as the conjectural lower bound, while the general conjecture is not presented as resolved.