Complete characterization of the tropical multiple Horn solution set for large networks

Let ${\rm mHorn}^{\Pi_1,\Pi_2,\Pi_3}$ be the set of 6-tuples realized by maximal multi-paths in the concatenation of planar networks $\Pi_1$, $\Pi_2$, and $\Pi_3$. The trace equalities, rhombus inequalities, and tetrahedron equalities are the conditions described for this set; in particular, if $m_\lambda,\dots,m_\tau$ are the numbers assigned to the relevant tetrahedron vertices, then

$$\alpha=m_\lambda+m_\nu,\qquad \beta=m_\mu+m_\tau,\qquad \gamma=m_\rho+m_\sigma$$

and the tetrahedron equalities are

$$\alpha\leqslant\operatorname{max}\{\beta,\gamma\},\qquad \beta\leqslant\operatorname{max}\{\gamma,\alpha\},\qquad \gamma\leqslant\operatorname{max}\{\alpha,\beta\}.$$

Tropical multiple Horn conjecture. For sufficiently large networks $\Pi_1,\Pi_2,\Pi_3$, for example standard networks, these conditions are necessary and sufficient. Equivalently, ${\rm mHorn}^{\Pi_1,\Pi_2,\Pi_3}$ is completely described by the trace equalities, tetrahedron equalities, and rhombus inequalities.

The paper proves necessity of these conditions. Sufficiency is conjectured for large networks; the standard-network setting is highlighted as an example but is not proved in the stated passage.