Span of critical points and irreducibility of structured low-rank varieties

Let $S e\varnothing$ be a zero pattern, let $U\in\mathbb{R}^{m\times n}$, and let $r\in[m-1]$. Denote by $Z_{U,r}^{S}$ the critical-point variety and by $H_U^{S}$ the corresponding affine linear space, and let $\mathcal{L}_r^S$ be the structured variety of matrices of rank at most $r$ with zeros prescribed by $S$. Span–irreducibility conjecture.

$$\langle Z_{U,r}^{S}\rangle=H_U^{S}\quad\Longleftrightarrow\quad\mathcal{L}_r^S\text{ is irreducible}.$$

This conjecture predicts that the linear span of the critical points recovers the natural affine space exactly when the structured low-rank variety is irreducible. The supplied text gives no evidence that the equivalence is resolved.