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

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 mean distance to the Dirichlet set by

$$\rho(\mathcal{G};{\mathsf{V}^{\mathrm D}}):=\frac{1}{|\mathcal{G}|}\int_\mathcal{G}\operatorname{dist}(x,{\mathsf{V}^{\mathrm D}})\operatorname{d}\!x.$$

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

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

for all unilateral metric graphs. This conjecture proposes that the first eigenvalue cannot be uniformly bounded above using only the mean distance to the Dirichlet set. The surrounding discussion notes that lower bounds involving mean distance are known, while the corresponding proposed upper bound is expected to fail.