Doubly connected graphs with arbitrarily close hot spots conjecture

Let $\epsilon>0$. A doubly connected metric graph is a graph of the type specified in the paper's location discussion. Let $\operatorname{diam}\gamma$ denote its diameter, $\mu_2(\gamma)$ its second eigenvalue, and $M$ its hot-spot set.

Close hot-spots conjecture. For every $\epsilon>0$, there exists a doubly connected graph $\gamma$ such that

$$\operatorname{diam}\gamma=1,$$

$mu_2(\gamma)$ is simple, and

$${\text{\rm max}}\{\operatorname{dist}(x,y):x,y\in M\}<\epsilon.$$

This shows that even without a tree-like boundary structure, the hot spots may be arbitrarily close relative to the graph diameter. The source gives no resolution status.