Affine relation for critical points with an admissible column subset

Assume that $\mathcal{L}_r^S$ is irreducible. Let $I\subset[n]$ satisfy $|I|=m$, and let $X_{[m],I}$ and $U_{[m],I}$ denote the corresponding $m\times m$ submatrices; write $C(U_{[m],I})$ for the cofactor matrix. Affine-relation conjecture. The complex critical points of $d_U$ on $\mathcal{L}_r^S$ satisfy the additional affine relation

$$\left\langle X_{[m],I},C(U_{[m],I})\right\rangle_F-r\det(U_{[m],I})=0$$

if and only if $S\cap([m]\times I)=\varnothing$. The conjecture characterizes precisely when this affine relation survives the imposed zero pattern; no resolution is given in the supplied text.