This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here.

Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be their von Neumann algebra tensor product.

Question:Can every projection in $M\bar{\otimes}N$ be expressed as the supremum (join) of projections of the form $p\otimes q$, where $p$ and $q$ are projections in $M$ and $N$, respectively?