Generic two-hot-spots conjecture for quantum graphs

Let $amma$ be a metric graph satisfying the standing assumptions of the paper. Write $mu_2(amma)$ for its second eigenvalue, and let $M$ be the set of global minima and maxima of corresponding second eigenfunctions.

Generic two-hot-spots conjecture. Generically, $mu_2(amma)$ is simple and the corresponding eigenfunction has exactly one minimum and one maximum, so that $|M|=2$.

The paper has results showing that this behaviour can be obtained after small modifications, but the generic assertion remains conjectural in the supplied text.