The absence of a universal upper bound for the first eigenvalue in terms of inradius

Let $\mathcal{G}$ be a unilateral metric graph with Dirichlet set ${\mathsf{V}^{\mathrm D}}$, let $\lambda_1(\mathcal{G};{\mathsf{V}^{\mathrm D}})$ denote its first eigenvalue, and define the inradius relative to the Dirichlet set by

$$\operatorname{inr}(\mathcal{G};{\mathsf{V}^{\mathrm D}}):=\max_{x\in\mathcal{G}}\operatorname{dist}(x,{\mathsf{V}^{\mathrm D}}).$$

Inradius upper-bound conjecture. There is no universal constant $C>0$ such that

$$\lambda_1(\mathcal{G};{\mathsf{V}^{\mathrm D}})\leq \frac{C}{\operatorname{inr}(\mathcal{G};{\mathsf{V}^{\mathrm D}})^2}$$

for all unilateral metric graphs. This is presented as a weaker version of the mean-distance conjecture. The source explains that a uniform lower bound in terms of inradius alone cannot hold, while the proposed uniform upper bound is left as an open conjecture.