Pumpkin hot-spots diameter conjecture

Let $\gamma$ be a non-equilateral pumpkin metric graph, and suppose that its second eigenvalue $mu_2(\gamma)$ is simple. Let $M$ be the set of global minima and maxima of the corresponding second eigenfunction, and let $\operatorname{dist}$ and $\operatorname{diam}$ denote graph distance and diameter.

Pumpkin hot-spots conjecture. The set $M$ has exactly two points and

$${\text{\rm max}}\{\operatorname{dist}(x,y):x,y\in M\}=\operatorname{diam}\gamma.$$

The claim formalizes the expectation that, for a non-equilateral pumpkin, the global extrema lie at the midpoints of the two longest edges and realise the diameter. The source gives no resolution status.