Let $\mathbb{P}^N$ be the projective space parametrizing $n\times n$ non-zero matrices modulo scalar multiplication, and let $\mathbb{P}^M\subset\mathbb{P}^N$ be the subspaces of symmetric matrices.

In $\mathbb{P}^N$ we have a stratification $X_1\subset X_2\subset \dots \subset X_r \subset \dots \subset \mathbb{P}^N$ where a general point of $X_r$ corresponds to a rank $r$ matrix. Such stratification restricts to the corresponding stratification $Y_r = X_r\cap \mathbb{P}^M$ of the space of symmetric matrices.

Assume there exists a polynomial of degree $d$ on $\mathbb{P}^M$ having multiplicity $m_r$ along $Y_r$ for all $r$. Can we affirm then that there exists a polynomial of degree $d$ on $\mathbb{P}^N$ having multiplicity $m_r$ along $X_r$ for all $r$ ?