Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$.

Assume the strong operator topology on $B(X)$ is separable and cardinal number of $B(X)$ is of the continuum.

Question. Can we conclude the norm topology on $X$ is separable?

Remark. It was proved by Tomek Kania that the assumption $|B(X)|=\mathfrak{c}$ is not solely enough to get the separability assertion.