Transition-system conjecture for symmetric matrix Schubert varieties

Let $n$ be a positive integer, let $\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}$ be the space of symmetric $n\times n$ matrices, and for $w\in\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}$ let $\mathsf{MSV}^{\mathrm{sym}}_w$ be the corresponding symmetric matrix Schubert variety, with ideals $I(\mathsf{MSV}^{\mathrm{sym}}_w)$ and $J^{\mathrm{sym}}_w$. Transition-system conjecture. There exists a transition system containing

$$\left\{\left(I(\mathsf{MSV}^{\mathrm{sym}}_w),J^{\mathrm{sym}}_w\right):w\in\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}\right\}.$$

The naive family is too small because it need not even be a partial transition system, and non-radical ideals are required. The conjecture is verified computationally for $n\leq4$, while its validity in general is open.