Let $X$ be a complex manifold and let $\mathcal{O}^*$ be the sheaf nonvanishing holomorphic functions on it. Does it have an acyclic cover? That is, a cover for which all open sets and all intersections are acyclic for this sheaf.

According to this related question, the sheaf $\mathcal{O}$ does since any complex manifold can be covered by Stein manifolds: https://math.stackexchange.com/questions/1232364/does-a-complex-manifold-always-admit-an-acyclic-cover-for-the-sheaf-of-holomorph

Therefore, the result would follow if the open sets and all intersections of this cover by Stein manifolds could be taken to be contractible, due of the long exact sequence in cohomology associated to the short exact sequence

$0\to \mathbb{Z}\to \mathcal{O}\to \mathcal{O}^*\to 0\,.$

But I don't know if this can be done.