This is a follow-up question after this

The set-up is almost the same as before,

Let $k$ be a number field, $p$ be a rational prime. Let $A$ be an abelian variety over $k$ which has a good reduction at all primes $\mathfrak{p}\subset k$ lying over $p$.

Suppose also that $k(A[p])\neq k$. Then

Does there exist a prime $\mathfrak{p}\subset k$ lying over $p$ such that $\mathfrak{p}$ is ramified in $k(A[p])$?

An easy case is when $\zeta_p=\exp(\frac{2\pi i}{p})\notin k$, since in this case, $p$ is totally ramified in $\mathbb{Q}(\zeta_p)$.

So, I am interested in the case when $\zeta_p\in k\neq k(A[p])$.