Generic non-optimality of constant potentials on non-path metric trees

Fix an underlying discrete tree graph that is not a path graph. Let $\mathcal{E}$ be its edge set, let the positive edge-length vector lie in $\mathbb{R}^{|\mathcal{E}|}_{+}$, and let $M>0$. For the corresponding metric tree $T$, let $C_{T,M}$ denote the class of convex potentials under consideration, and let the spectral gap be the difference between the first two eigenvalues.

Generic non-optimality conjecture. The set of edge-length vectors and values $M>0$ for which the constant potential minimizes the spectral gap in $C_{T,M}$ is of the first Baire category in

$$\mathbb{R}^{|\mathcal{E}|}_{+}\times\mathbb{R}_{+}.$$

The claim concerns the generic behavior of convex-potential gap minimization on metric trees and asserts that constant potentials are exceptional for every underlying tree that is not a path graph. The source presents it as a conjecture; no resolution is supplied.