Irreducibility conjecture for the rank varieties of the binary tree model

Let $T$ be a binary tree, let $\mathcal{M}_T$ be its general Markov model, and consider the rank varieties described in (1) and (2): the varieties defined by the rank conditions on the selected pendant-edge slices and on the matrix flattenings compatible with $T[e]$ but not with $T$. Irreducibility conjecture. The rank varieties in (1) and (2) are irreducible. At present, it is unknown whether these equational descriptions suffice to cut out the codimension-one subvarieties $\overline{\phi(F)}$ of $\mathcal{M}_T$; proving irreducibility of the rank varieties would suffice.