While it is possible to argue that "all pigs in play" is more specific than "all pigs," it is definitely less specific than a single target pig. And, since the rule says "the more specific of the two rules takes precedence," (emphasis added) it is pretty hard to argue that Primal isn't "more" specific than Pig Sling.
Thus, because both rules are constantly in conflict until the Primal condition ends, I think the "more specific" rule pretty much answers the question. Primal takes precedence.
Thanks to everyone for helping sort through this.