Let $u:A \to \prod_{\mathcal U} M_n$ be a unital completely positive map (ucp) from a unital separable $C^*$algebra into the von Neumann algebra ultraprodut $\prod_{\mathcal U} M_n$.

Here $\mathcal U$ is an ultrafilter on $\mathbb N$ and $\prod_{\mathcal U} M_n$ is the quotient of $B=\{(x_n)_{n \in \mathbb N}, x_n \in M_n(\mathbb C) , \sup_n \|x_n\|<\infty\}$ by the ideal $I_{\mathcal U}=\{(x_n)_n, \lim_{\mathcal U} Tr(x_n^* x_n)/n = 0\}$.

Does there exist a ucp lifting of $u$, i.e. a sequence $u_n:A \to M_n(\mathbb C)$ of ucp maps such that $u=q \circ (u_n)$, where $q:B\to B/I_{\mathcal U}$ is the quotient map?

If not, is $u$ locally liftable? That is: given a finite dimensional operator system $E\subset A$ (= a subspace $E$ of $A$ containing $1$ and stable under $a \mapsto a^*$), does there exist a ucp lifting of the restriction of $u$ to $E$?

Some comments: as in my related question, I know that the answer is no in general if one replaces the von Neumann algebra ultraproduct by the $C^*$-algebra ultraproduct. But I hope that again, the situation might be much simpler in the von Neumann algebra setting. (I even have the feeling that I have known the answer to this question, but that I have forgotten it).