Depends on what our definition of "best" is
The most nationally followed and cared about rivalry is UNC and Duke
The most hatred between the two fanbases is KU and Missouri
U of L and UK are a mix of the two. Could be better than what it is if U of L actually won more.
I don't think win/loss plays that much of a role, tbh....Duke/UNC are basically even, but the rivalry is nowhere near as intense as UL/UK. I mean even IF SOME UK fans say the game is "meh" due to how lopsided it has been, you can bet your ass those same fans wanna win---Regardless how one-sided it has been.
Also, I really think you are overlooking the Indiana/UK rivalry-----and even more so, the Indiana/Purdue. But I 100% agree with KU/Mizzou. That shit goes beyond sports.
Note: I've mentioned this already, but----Cincy/Xavier is ****ing ridiculous. About 6 years ago or so---I think---I worked UC/Xavier BASEBALL game. Dude.........Yikes. The shit yelled from the stands, and the dugouts----Wow. Not sure what inning, but a cat from Xavier hit a ****ing bomb----He gave it a nice looooooooon look, before flipping his bat, and yelling---"Little League fastball". Welp next dude caught one in the back, abd well.............Here we go. Warnings issued, but no ejections, and really no further issues. But ****, you could feel the intesity, the hate---and were just waitng for a brawl.
Those two schools ****ing hat one another...