I think in terms of the church the term “conservative” does not necessarily align with the political understanding. The largest shift in the church is demographic. White Europeans are loosing power rapidly in the catholic church. Subsequently the catholic church will align less and less with “Western” conservatives, and their imperialist understanding of geopolitics.

I expect to see much more opposition of “christian conservative” politicians in Europe towards the catholic church.
Ironically just yesterday the president of the German federal parliament of the “christian conservative” CDU told the Churches to shut up about politics.