Generic small-time null-controllability conjecture for the heat equation on graphs

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 $(\ell_i)_{i=1,\dots,N}$.

Generic heat-control conjecture. Generically with respect to the lengths $(\ell_i)_{i=1,\dots,N}$, the heat equation is small-time null-controllable.

The claim is presented as a natural extension of the stated result for generic star graphs, where small-time null-controllability holds for every $T>0$. Its validity for more general graph structures remains open.