In a preliminary version of what would become Gromov's "A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature", he writes on page 88:
Neither is one able to prove (or disprove) that manifolds with positive scalar
curvatures have zero simplicial volumes.
Possibly, these conjectures need significant modifications to become realistic.
Simplicial volume didn't make it into the published version of these notes, maybe because he thought his conjectural relationship between scalar curvature and simplicial volume was too hopeless to be worth mentioning, but at any rate this is reasonable evidence that this question was an open problem in 2017, and I haven't seen any evidence of progress since then.