The generating conjecture for homogeneous invariants of the 2-state GM+I model

Let $T$ be an $n$-taxon binary tree, and let $p_{11\dots1}$ and $p_{22\dots2}$ denote the joint-distribution coordinates in which every state is respectively $1$ and $2$. Consider the edge flattenings of the joint-distribution tensor, their $3\times 3$ minors, and the stochastic invariant for the 2-state general Markov model with invariable sites (GM+I).

Generating conjecture. The ideal of homogeneous invariants for the 2-state GM+I model on $T$ is generated by those $3\times 3$ minors of edge flattenings that do not involve the variables $p_{11\dots1}$ and $p_{22\dots2}$, together with the stochastic invariant.

This gives a proposed complete generating set for the homogeneous phylogenetic ideal of the 2-state GM+I model on every binary tree, extending the known edge-flattening invariants while accounting for the two all-equal-state coordinates.