The zero simplicial volume characterization for nonpositively curved four-manifolds

Let $M$ be a closed nonpositively curved $4$-manifold, with simplicial volume $||M||$ and Euler characteristic $\chi(M)$. Zero simplicial volume characterization conjecture. Then

$$||M||=0 \quad\Longleftrightarrow\quad \chi(M)=0.$$

This conjecture is stronger than the asserted implication from zero simplicial volume to vanishing Euler characteristic in the nonpositively curved four-dimensional setting. The converse is known for surfaces and false in dimensions at least six, while the source says that the dimension-$4$ case remains mysterious; hence the conjecture is open.