Generic graph geometric control conjecture for the wave equation

Let $\mathcal{G}$ be a graph with $N$ edges $(e_i)_{i=1,\dots,N}$ and let $\omega$ be an open subset of $\mathcal{G}$. The lengths of the edges are $(\ell_i)_{i=1,\dots,N}$, and (GGCC) denotes the graph geometric control condition.

Generic wave control conjecture. Generically with respect to the lengths $(\ell_i)_{i=1,\dots,N}$, (GGCC) is a necessary and sufficient condition for the internal control of the wave equation, whatever the boundary conditions are.

The paper explains that (GGCC) can fail to be necessary for specially chosen graphs and boundary conditions, while conjecturing necessity and sufficiency for generic edge lengths. The general assertion remains open.