Knutson's scheme-theoretic equations conjecture for upper-upper schemes

Let $X$ and $Y$ be pairs of $n\times n$ matrices, let $\pi\in S_n$, and let $P_\pi$ be the permutation matrix of $\pi$, with transpose $P_\pi^{\mathsf T}$. Let $\overline{E}_\pi$ denote the closure of the corresponding subset of the upper-upper scheme. For each pair $i,j$, consider the lower-left $i\times j$ rectangle in $X$ and in $Y$.

Knutson's conjecture. The variety $\overline{E}_\pi$ is defined as a scheme by the following three sets of equations and rank conditions: $XY$ and $YX$ are upper triangular;

$$\operatorname{diag}(XY)=\operatorname{diag}(P_\pi YX P_\pi^{\mathsf T});$$

the rank of the lower-left $i\times j$ rectangle in $X$ (respectively, in $Y$) is bounded above by the number of $1$s in the corresponding rectangle of $P_\pi$ (respectively, of $P_\pi^{\mathsf T}$), giving the equations defining the associated matrix Schubert varieties.

This is attributed in the source to Knutson and concerns a scheme-theoretic description of closures of strata in the upper-upper scheme. The source gives the conjecture but no evidence of resolution.