Let $X\subseteq B(H)$ be an operator system and let $M\subseteq B(K)$ be a von Neumann algebra. We form the Fubini-tensor product
$$X \otimes_\mathcal{F} M := \{z \in B(H\otimes K): (\sigma\otimes \iota)(z) \in M \text{ and } (\iota \otimes \tau)(z)\in X \text{ for all }\sigma\in B(H)_*, \tau \in B(K)_*\}.$$
We have a natural injective linear map
$$X \otimes_\mathcal{F}M \to B(M_*,X): z \mapsto (\omega \mapsto (\iota \otimes \omega)(z)).$$

Is it true that the image of this map consists of all completely bounded maps $M_*\to X$? 

Note that this is at least true if $X$ itself is a von Neumann algebra (However, this result is not very easy to find in the literature).