To me it seems more and more likely that the battle for Laugh Tale would be a four way fight. With Law being used to most likely hype and setup BB and Kid for Shanks, I find it hard to believe that those two (BB and Shanks) will not be meeting each other near/at Laugh Tale. Shanks was pretty confident when he said that they were going for the OP. Being a former member of Roger Pirates he already probably knows where the RPs are located or he already has been able to get the rubbings. BB also has the RPs apparently, the reason I think they kidnapped Pudding for.
There were six candidates for the OP after Wano:
1. Red Hair Pirates (unknown RPs, possibly all four, alliance with Elbaf probably means they got someone to translate like Saul)
2. Blackbeard Pirates (unknown RPs, have pudding)
3. SHPs (three RPs, have Robin)
4. Heart Pirates (two/three RPs, WCI RP is unclear, no translator though, going for the man with the burn mark)
5. Kid Pirates (two/three RPs, Zou RP is unclear, no translator though, going for the man with the burn mark)
6. Cross Guild (have Croc who probably has some idea about it)
With Law vs BB and now Kid vs Shanks most likely to happen, it appears out of the 6 there will be two dropouts (Kid and Law, serving as hype tools). That leaves RHPs, BBPs and SHPs as the real contenders. Cross Guild is the wildcard here, we know Buggy wants/dreams to be PK and so far he has become a Warlord and now Yonko (gag means), but they also have Crocodile who most likely knows about the RPs (Alabasta search for Pluton through Poneglyphs). And while the crew itself seems weak (its way stronger than Kid and Law pirates though), I think we can expect them to also be able to recruit some real big names after Mihawk and Crocodile are shown as members.
Of course there are other factions like WG, Navy and WGs which might also be involved.