Conjecture on simultaneous resolution of related tropical conifold nodes

Let $(B,\mathcal P,\phi)$ be a tropical conifold with nodes $p_1,\ldots,p_k$. The nodes are related if there exist tropical $2$-cycles $(S_1,j_1),\ldots,(S_r,j_r)$ such that $\bigcup_{m=1}^r j_m(S_m)$ contains precisely the nodes $p_1,\ldots,p_k$. Simultaneous resolution means resolution by the subdivision, compatible fan structure, and strictly convex MPL-function described in the source.

Related-nodes conjecture. A set of nodes $p_1,\ldots,p_k$ in $(B,\mathcal P,\phi)$ can be simultaneously resolved by the above process if and only if they are related.

The claim gives a necessary-and-sufficient tropical-cycle criterion for simultaneous resolution. The paper presents it as a conjecture; no resolution status is supplied here.