Groebner-basis conjecture for symmetric matrix Schubert ideals

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 let $I^{\mathrm{sym}}_w$ and $J^{\mathrm{sym}}_w$ be the minor ideal and antidiagonal ideal associated to a symmetric matrix $w\in\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}$. Groebner-basis conjecture. For every such $w$,

$$\operatorname{init}(I^{\mathrm{sym}}_w)=J^{\mathrm{sym}}_w,$$

and the minors generating $I^{\mathrm{sym}}_w$ form a Groebner basis in the reverse lexicographic term order. This would give explicit Groebner bases and initial ideals for the symmetric matrix Schubert ideals. It is verified computationally for $n\leq7$ over any field, but remains open in general.