Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $h$ be an $L^1(\mathbb{R}^n)$-valued integrable random variable that is jointly measurable, i.e. $h\in L^1(\mathbb{P}; L^1(\mathbb{R}^n))$ and $(\omega, x)\mapsto h(\omega)(x)$ is $\mathcal{F}\otimes \mathcal{B}^n$-measurable.
Let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. Does it hold for suitable $\mathbb{R}^n$-valued r.v.'s $x$ that $$\omega\mapsto E(h\vert \mathcal{G})(\omega)(x(\omega)) = E\big(\omega\mapsto h(\omega)(x(\omega))\big\vert \mathcal{G}\big)\quad \text{in $L^0(\mathbb{P})$} ?$$
Thanks!