Generation conjecture for sparse two-factor analysis model ideals

A sparse two-factor analysis model is the model associated with a graph $G$ having two latent variables, with $M_{\leq 1}(G)$ denoting the set of model polynomials indexed by subsets of variables of size at most one. Generation conjecture. The ideal of the sparse two-factor analysis model corresponding to graph $G$ is generated by off-diagonal $3\times3$-minors, pentads, and the polynomials in $M_{\leq 1}(G)$.

The conjecture is motivated by Gröbner-basis computations for sparse two-factor models with larger overlaps among the children sets of the latent variables. It proposes that these three families suffice to generate the full ideal, beyond the low-degree generators observed in the computed examples.