If $d$ and $n$ are positive integers, does there exist a constant $B=B(d,n)$ with the following property?

For any $n$-dimensional abelian variety $A$ over a degree-$d$ number field $K$, there is an extension $L/K$ with $[L:K]\le B$ such that $A(L)$ is Zariski-dense in $A$.

Remarks:

1. Of course, for $n=1$ we can take $B=2$. For arbitrary $n$, is it even possible to choose $B$ depending only on $n$?

2. This is reminiscent of this previous MO question.