Here's a question I've been thinking about, it's a curiosity that I don't know how to answer. There could be a simple counterexample, or it could be true and I don't know how difficult it would be to prove.

If we fix $m$, is it always possible to find a sufficiently large $n$ satisfying the conditions of the following question: [Note: My original question was to determine whether it is true for arbitrary $m,n$ which was answered below negatively; I have edited it to make the question more interesting].

Define $ \phi: S_m \rightarrow S_{m+n}$ is a canonical embedding, and $\phi^{*} : F[S_m] \rightarrow F[S_{m+n}]$ and similarly embeddings $\theta: S_{n} \rightarrow S_{m+n}$, and the induced map, such that $\phi(S_{m}) \times \theta(S_{n})$ is a direct product.
Given an element $x \in \phi^{*}(F[S_m]), x \neq 0$, is it necessary that there exist an element $x' \in F[S_{m+n}]$ so that the product $xx' \in \theta^{*}(F[S_n]), xx' \neq 0$. It seemed true in the cases that I have tried, but they are quite small so I'm not certain if this is true.

Making the assumption $ \text{char} F = 0$ would make it easier I'm sure, but even in this case I can't prove it.