Consider a set S of partitions not containing the empty partition (I would be happy with, say, all the partitions of size less than k, except for the empty one).

Let $U_\lambda^{(r)}$ be the zero-set in $\mathbb{C}^r$ of the Schur polynomial $s_\lambda(x_1,\cdots,x_r)$.

What is known about $\cap_{\lambda \in S} U_\lambda^{(r)}$, beyond the fact that it is symmetric under the action of $S_r$?

(I am having trouble finding information about this: all the hits are about the different question of the Schur stability of univariate polynomials, a concept based on the location of the roots of those polynomials).