Weak convergences in Bochner spaces

I'm having bit trouble in understanding weak convergences in Bochner space. I have following question for some general measurable space $$\Omega$$:

Let $$\{x_n\}$$ be a bounded sequence in $$L^2((0,T)\times \Omega ; H^1(\mathbb{R}^d))$$, then we can extract a weakly convergent sub sequence $$\{x_{n_k}\}$$ st $$x_{n_k} \rightharpoonup x$$ in $$L^2((0,T)\times \Omega ; H^1(\mathbb{R}^d))$$. Can we conclude for a.e. $$t$$, $$x_{n_k}(t) \rightharpoonup x(t)$$ in $$L^2(\Omega ; H^1(\mathbb{R}^d))$$?

(a) is not even true in ordinary Lebesgue spaces so it's certainly not true in Bochner. The standard counterexample is something like $$x_n(t) = e^{2\pi i n t}$$ in $$L^2([0,1])$$. It converges weakly to 0 but $$x_n(t)$$ diverges for almost every $$t$$.
In your specific context, you could take instead $$x_n(t) = e^{2 \pi i n t} \phi$$ where $$\phi$$ is some fixed nonzero element of $$L^2(\Omega; H^1(\mathbb{R}^d))$$.