Primality 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 for each symmetric matrix $w\in\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}$ let $I^{\mathrm{sym}}_w$ be the ideal generated by the specified minors in the coordinate ring $\mathbb{K}[\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}]$. Primality conjecture. For every symmetric matrix $w$, $I^{\mathrm{sym}}_w$ is a prime ideal of $\mathbb{K}[\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}]$. Since the corresponding symmetric matrix Schubert varieties are irreducible, this would follow from the ideal-equality conjecture. It is computationally verified for $n\leq5$ over $\mathbb{Q}$, $\mathbb{F}_2$, and $\mathbb{F}_3$, but remains open in general.