Minimal generator conjecture for the two-factor model

Let $I_{p,2}$ be the ideal of invariants of the two-factor model with $p$ observed variables. The off-diagonal $3\times3$-minors and pentads are the candidate generators.

Two-factor minimal-generation conjecture. The ideal $I_{p,2}$ is minimally generated by

$$5\binom{p}{6}$$

off-diagonal $3\times3$-minors and

$$\binom{p}{5}$$

pentads.

The conjecture is supported by the paper's numerical evidence. These generators involve at most six observed variables, but the source notes that they do not appear to form a Gröbner basis under any term order.