Linear-eliminant ideal-membership conjecture for factor-analysis invariants

Let $\Psi$ be the covariance matrix of a factor-analysis model. For $m\geq 2$, let $f_{i,R,C,\bar R,\bar C}$ be a linear eliminant, and let the off-diagonal $(m+1)\times(m+1)$-minors of $\Psi$ generate an ideal.

Linear-eliminant membership conjecture. If $m\geq 2$, $p\geq 2m+2$, and $R\cup C\neq\bar R\cup\bar C$, then

$$f_{i,R,C,\bar R,\bar C}$$

is in the ideal generated by the off-diagonal $(m+1)\times(m+1)$-minors of $\Psi$.

This is proposed as the converse to a preceding non-membership proposition. It forms part of the paper's broader finiteness conjectures for the ideals $I_{p,m}$, and the source provides no proof.