That is true, I saw Kaido was being pretty kind/sympathetic to Jack on the rooftop but that was it. Yeah that does seem true with King and Kaido actually, it would explain why he's not given a fuck about any of his Tobi Roppo and Calamities being beaten now. Ironically he inspired such loyalty from some of them too.
I really wish Oda had gone through with making the Tobi Roppo challenge the All Stars or whatever, so we could have delved more into their personalities, connections with each other and how Kaido views them too.