Cinkir's explicit tau lower bound conjecture for metrized graphs

Let $\Gamma$ be a metrized graph, let $\tau(\Gamma)$ denote its tau constant, and let $\ell(\Gamma)$ denote its total length. Cinkir's explicit tau lower bound conjecture. For every metrized graph $\Gamma$,

$$\tau(\Gamma)>\frac{1}{108}\cdot\ell(\Gamma).$$

This is presented as a refinement of Baker and Rumely's universal positive lower-bound conjecture. The paper describes $\frac{1}{108}$ as the conjectural lower bound and proves asymptotic convergence to this value for one family, the hexagonal nets around a torus, but does not establish the inequality for all metrized graphs.