They've done it with Man United, Chelsea and now City. 3 rivalries peaking at different times to a none local rival, the one constant being Liverpool fans. I wouldn't be surprised if there's been one with Arsenal at some point. That's because, it seems all it takes is for one club to be in the way of "their rightful perch" in their entitled eyes.
https://talksport.com/football/550782/most-trophies-england-liverpool-european-cup/
After reading the above Liverpool arse-kissing major honours in English football ranking(and discounting the inter cities fairs cup which is not a major honour for a number of reasons). I started doing a top 10 in terms of major honours as of 1960, since most Liverpool fans are adamant they were massive club way above City before then, when their Littlewoods Pools win comes up in conversation.
It took me longer than I expected to get 10 places because there were so many teams level, so it's not quite a top 10 but it's easy to get the gist of the landscape back then.
1960 Major trophies ranking in English football:
1: Aston Villa 13
2: Arsenal 10, Newcastle 10
3: Blackburn 8
4: Everton 7, Man United 7, Wolves 7, Sheffield Wed 7, Sunderland 7
5: West Brom 6
6: Liverpool 5, Sheffield United 5
7: Man City 4, Preston NE 4
8: Spurs 3
Chelsea 1
I did it again for 1976 where City moved to 6th but it's less relevant to Liverpool(2nd) so I wont bother with that.