Let $f\colon I \times X \to \mathbb{R}$ be a map where $I \subset \mathbb{R}$ is an interval, $X$ is a Banach space (possibly non-separable) and we have $$t \mapsto f(t,x) \text{ is measurable}$$ $$x \mapsto f(t,x) \text{ is continuous}.$$

My question is: given $w \in L^1(0,T;X)$, is $t \mapsto f(t,w(t))$ measurable or not?

When $X$ is a separable space, it is true. See this paper. I've seen also claims that this holds when $X$ is not separable, but all such claims have "proofs" cited to the above linked paper, which only handles the separable case. So is separability required or not?