The cherry bound for minimal support networks of HGT networks

Let $N$ be a level-$k$ HGT network with a biconnected component $B$. Let $N(B)$ denote the minimal support network of $B$, let $T_{N(B)}$ denote its underlying tree, and call a pair of leaves sharing a parent a cherry. A reticulation arc is an arc entering a reticulation vertex. Cherry-bound conjecture. The network $N(B)$ has at most $k-1$ cherries unless $N(B)$ strictly embeds in the top of $N$ or all reticulation arcs in $N(B)$ are between adjacent arcs of $T_{N(B)}$. This conjecture concerns how much reticulation complexity can be concealed while a network remains tree-metrizable. Its resolution could clarify the structures underlying tree-metrizability; the source gives no evidence that it has been proved or refuted.