Universal positive lower bound for the tau constant of metrized graphs

Let $\Gamma$ be a metrized graph, with total length $\ell(\Gamma)$ and tau constant $\tau(\Gamma)$. The ratio $\tau(\Gamma)/\ell(\Gamma)$ is invariant under scaling all edge lengths by the same positive constant. Universal lower-bound conjecture. There is a universal constant $C>0$ such that, for all metrized graphs $\Gamma$,

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

The conjecture asks for a positive scale-independent lower bound for the tau constant. The source notes that the conjecture was subsequently proved by Zubeyir Cinkir, with $C=1/108$.