Minimal generators for symmetric ideals

Let $R = K[X]$ be the polynomial ring in infinitely many indeterminates $X$ over a field $K$, and let ${\mathfrak{S}}_{X}$ be the symmetric group of $X$. The group ${\mathfrak{S}}_{X}$ acts naturally on $R$, and this in turn gives $R$ the structure of a module over the group ring $R[{\mathfrak{S}}_{X}]$. A recent theorem of Aschenbrenner and Hillar states that the module $R$ is Noetherian. We address whether submodules of $R$ can have any number of minimal generators, answering this question positively.