Ideal and initial-ideal equality 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 $I^{\mathrm{sym}}_w$ be the ideal generated by the specified minors, $\mathsf{MSV}^{\mathrm{sym}}_w$ the corresponding symmetric matrix Schubert variety, and $J^{\mathrm{sym}}_w$ the specified antidiagonal ideal. Ideal-equality conjecture. For every $w\in\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}$,

$$I^{\mathrm{sym}}_w=I(\mathsf{MSV}^{\mathrm{sym}}_w)\quad\text{and}\quad J^{\mathrm{sym}}_w=\operatorname{init}\left(I(\mathsf{MSV}^{\mathrm{sym}}_w)\right).$$

The equalities would identify the determinantal equations and their initial ideals for symmetric matrix Schubert varieties. They are consequences of the proposed transition-system construction and remain open in general.