Let $k$ be a field of characteristic 0, and $\overline k$ be a fixed algebraic closure of $k$. Let $k\subset F\subset E$ be a tower of finite Galois extensions in $\overline k$, where both $\mathrm{Gal}(F/k)$ and $\mathrm{Gal}(E/F)$ are nonabelian simple finite groups. We have a norm map $N\colon E\to F$. Choose $x\in F^*$ which is not a norm, i.e. is not contained in the image of $N$.

Now let $K/k$ be a finite solvable field extension in $\overline k$. Then $K\cap E=k$, and we have a norm map of composites $N_K\colon KE\to KF$.

Does there always exist a finite solvable field extension $K/k$ such that $x$ is contained in the image of the map $N_K$?

If $k$ is a number field, the answer is YES. For an arbitrary field $k$ I expect in general the answer NO, but I cannot construct a counter-example.