Let $H$ be a Hilbert space, and

let $A_1,A_2,A_3\subset B(H)$ be three commuting von Neumann algebras.

We write $\odot$ for the algebraic tensor product, and $\bar\otimes$ for the spatial tensor product of von Neumann algebras.

Suppose that for every $i,j\in\{1,2,3\}$, the map $A_i\odot A_j\to B(H)$ extends to a map $A_i\,\bar\otimes\, A_j\to B(H)$.

Does it follow that the map $A_1\odot A_2\odot A_3\to B(H)$ extends to a map $A_1\,\bar\otimes\, A_2\,\bar\otimes\, A_3\to B(H)$?