Sorry, this may only be a matter of words, but: I would not say “It is necessary to not adjust for multiplicity to get a Bayesian-frequentist equivalence”. The way I’d put it, a Bayesian method does not “adjust for multiplicity” because the account for multiplicity is pre-loaded (as it were) in its framing of the question and model to address what a frequentist calls a “multiplicity problem”. As per their culture, frequentists differ only in allowing a zoo of incompletely specified questions and models, an issue that Leamer dealt with at length in his 1978 book. If we demand the same degree of specification for frequentists as typical Bayesians do as per their culture, and also extend Bayesian methods to their closure (including limits approaching improper priors) then as per Wald 1947-50 we can map all admissible frequentist methods to Bayesian ones and back again. We can also map between them using hierarchical embeddings as per Good (The American Statistician 1987, Vol. 41, No. 1, p. 92).
See my general response a moment ago and the 2019 and 2021 cites therein, and also earlier treatments for Bayes in multiplicity settings such as these two:
and