If the relative norm and waek topologies of $B_X$ conincide at a point then the relative weak-$^\ast$ and norm topologies coincide in $B_{X^{**}}$

Question

Let $X$ be a Banach space and let the relative norm and weak topologies coincide at a point $x\in B_X$. That means that every open neighborhood of $x$ in the norm topology contains an open neighborhood of $x$ in the weak topology, and visa versa.

Does it follow that the relative norm and the weak-$^\ast$ topologies of $B_{X^{**}}$ coincide at $j(x)\in B_{X^{**}}$, where $j$ is the natural isometry to the double dual.

p.s. I need the above to answer this question.