Explicit-generator conjecture for squared Schubert varieties

For $1\leq i<j\leq n$, let $\mathcal{S}_{ij}$ be the Schubert variety in ${\rm Gr}(2,n)$, let $\mathcal{S}_{ij}^2$ be its square image, let $P$ be the associated symmetric matrix, and let $P_{rs}$ denote the $2\times n$ submatrix of $P$ given by rows $r$ and $s$. Explicit-generator conjecture. The prime ideal of $\mathcal{S}_{ij}^2$ is generated by the entries of

$$2P^2-\operatorname{trace}(P)\cdot P,$$

the $3\times3$ minors of $P$, and the $2\times2$ minors of the submatrices $P_{rs}$ where $r<i$ or $s<j$. This is the precise proposed generating set; its general validity remains unproved in the supplied text.