I don't see any other way for the samurai if they don't claim a death for Kaido, now he's defeated in the hole, but assuming he doesn't react and is captured, nothing is more natural than the people who are tormented by the misfortune that befell them. the Oden that was executed, wanting Kaido to be too, isn't it?
Zoro at the beginning of the arc was labeled as an assassin and they were already going to execute him, why wouldn't they execute Kaido? Oden was executed, shouldn't the people want an "an eye for an eye, a tooth for a tooth"?
Kaido's chances of being executed are high, in my opinion.