Many papers on Bayesian persuasion adopt SPE as a solution, by arguing that the sender has no private information. This does not make sense to me: Regardless of whether the sender has private information, we need to restrict the belief of the receiver about the state of the world after the receive observes an off-path disclosure policy, which SPE has no bite. Should we understand SPE in Bayesian persuasion literature as something that does not really mean subgame perfect equilibrium?
Here are my thoughts. Because the sender commits to a disclosure policy, how you stipulate off-path beliefs doesn't matter (phrased loosely). That is, the modification to PBE is trivial: any optimal persuasion policy corresponds to a class of optimal persuasion policies in which any belief is assigned to each off-path message.
Moreover, you could also resolve your off-path message beliefs issue by assuming that the cardinality of the message set is so small that no off-path messages are sent. This would correspond to, e.g., there only being two messages available in the canonical prosecutor-judge example from KG.