Rooted-triple reconstruction conjecture for binary normal networks

Let $X$ be the leaf set, and let $N_1$ and $N_2$ be binary normal networks on $X$ with no near-sibling reticulations. Write $R(N)$ for the set of rooted triples displayed by a network $N$, and write $N_1\cong N_2$ when the networks are isomorphic. Rooted-triple reconstruction conjecture. If $N_1$ and $N_2$ have no near-sibling reticulations, then

$$R(N_1)=R(N_2)\quad\Longleftrightarrow\quad N_1\cong N_2.$$

The paper's main theorem proves this reconstruction statement under the stronger assumption that both networks have no near reticulations, while examples show that excluding near-sibling reticulations is necessary in general. Whether the weaker hypothesis also suffices when near-stack reticulations are allowed remains open.