Involution pipe-dream decomposition conjecture for symmetric initial ideals

Let $n$ be a positive integer, let $I_\infty^{(n)}$ be the set of involutions indexing symmetric $n\times n$ partial permutation matrices, and let $J^{\mathrm{sym}}_y$ be the associated ideal for $y\in I_\infty^{(n)}$. For an involution pipe dream $D$ for $z\in I_\infty^{(n)}$, let $m_D(i,j)\in\{1,2\}$ be its crossing multiplicity and define

$$J^{\mathrm{sym}}_D:=\left\langle u_{ij}^{m_D(i,j)}:(i,j)\in D\right\rangle.$$

Involution pipe-dream conjecture. If $y\in I_\infty^{(n)}$, then

$$J^{\mathrm{sym}}_y=\bigcap_D J^{\mathrm{sym}}_D,$$

where $D$ ranges over all involution pipe dreams for all $z\in I_\infty^{(n)}$ with $z\geq y$ in Bruhat order. This would provide a primary decomposition of the symmetric initial ideals analogous to known decompositions for ordinary and skew-symmetric matrix Schubert varieties. The statement is supported by computations, but its general validity remains open.