Let $X$ and $Y$ be Banach spaces and let $\mathcal{B}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. As noted in the answers to a question on

https://math.stackexchange.com/questions/535645/weak-operator-topology-and-finite-rank-operators

the finite rank operators lie dense in $\mathcal{B}(X,Y)$ with respect to the strong operator topology (SOT-topology). I would like to know under which assumptions on the Banach spaces $X$ and $Y$ one can approximate an arbitrary $T\in\mathcal{B}(X,Y)$ in the SOT-topology by a *norm-bounded sequence* of finite rank operators.

It seems to me that the proof of denseness of the finite rank operators cited above does not yield approximation by norm-bounded sequences. I can obtain the required approximation under the assumption that $X$ is separable and that either $X$ or $Y$ has the bounded approximation property. However, these assumptions yield a stronger statement, namely that one can approximate a general $T\in\mathcal{B}(X,Y)$ by a norm-bounded sequence of finite-rank operators in the topology of uniform convergence on compact subsets of $X$. This clearly implies SOT-convergence, but it might be a strictly stronger statement than what I am looking for.

Does anyone know whether one can approximate an arbitrary $T\in\mathcal{B}(X,Y)$ by a norm-bounded sequence of finite-rank operators in the SOT-topology for more general Banach spaces than those mentioned above?