Homologically area-minimizing surfaces mod $v$ have at worst codimension 2 singular sets asymptotically

De Lellis and coauthors have proved a sharp regularity theorem for area-minimizing currents in finite coefficient homology. They prove that area-minimizing mod $v$ currents are smooth outside of a singular set of codimension at least $1.$ Classical examples like triple junctions demonstrate that their result is sharp. Surprisingly, even though their regularity theorem cannot be improved for any fixed $v$, if one instead fixes the homology class, then $v$ asymptotically one can always achieve more regularity. For any integral homology class $[\Sigma]$ on any Riemannian manifold, we show that for $v$ large, any area-minimizing mod $v$ current in $[\Sigma\mod v]$ must be an integral current, thus having a singular set of codimension at least $2$ in general and of codimension at least $7$ in the hypersurface case. Similar results are obtained for Plateau problems in Euclidean space. Our work is inspired by Morgan's work and based on De Lellis' and coauthors' work.