Generic graph geometric control conjecture for the Schrödinger 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 Schrödinger 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 Schrödinger equation.

The statement is motivated by the known necessity and sufficiency result for generic star graphs and by the existence of exceptional configurations where (GGCC) is not necessary. Its extension to almost all graphs remains open.