My question deals with a version of Arveson's extension theorem (for the standard version, see, e.g., Paulsen's book Completely Bounded Maps and Operator Algebras). Let $\mathcal A$ be a von Neumann algebra, $\mathcal R\subset\mathcal A$ an operator system, and $\mathcal H$ a Hilbert space. If $\Phi_0:\mathcal R\to\mathcal L(\mathcal H)$ is a linear normal completely positive unital map, does there exist a linear normal completely positive unital extension $\Phi:\mathcal A\to\mathcal L(\mathcal H)$ for $\Phi_0$ (complete positivity for a linear map on an operator system defined as in Paulsen's book)?
I am aware that if one gives certain restrictions for the operator system, the extension can be carried out. E.g., when $\mathcal A$ is a type-I factor and $\mathcal R$ is contained in the ultraweak closure of the set of its compact operators, the Arveson-like extension result for normal maps holds. Does anyone know whether the normal extension exists in general or whether there are some weaker versions of Arveson's theorem for normal maps? I am mostly interested in the case where $\mathcal A$ is a type-I factor. Thank you in advance.