The consistency-strength conjecture for ZFR with dependent choice

Let $cmathsf{ZFR}$ denote the set-theoretic theory under discussion, and let $cmathsf{DC}$ denote the principle of dependent choice. Consistency-strength conjecture. The theory $cmathsf{ZFR}+cmathsf{DC}$ has consistency strength strictly greater than that of $cmathsf{ZFR}$. This is motivated by the interaction between Berkeley-cardinal principles, failures of dependent choice, and elementary self-embeddings of the universe; the source does not provide a resolution.