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Just to clarify... the "NuGet v3 API" is intended for internal-use (i.e. only nuget.exe and NuGet VS extension). Microsoft's new position is that they do not intend on making this API public or stable in the foreseeable future, and that they will continue to support the "NuGet v2 API" (which is documented and public) indefinitely. They actually use ProGet to test changes to v2 API.
That said, until this position changes, we will not reverse-engineer the v3 protocol. So this means you will need to use nuget.org/api/v2 in your connector, not the v3 url.
Both v2 and v3 API work with client tools, both support DotNetCore, etc. If you're finding a issues with DNC projects, then it's likely related to framework monikers or something like that. Please submit a ticket for this with specifics, so we can investigate and patch.