Interpolational Craig property in logics of proofs with strong provability operator (Q1275992)

In a paper by \textit{E. Yu. Nogina} [Logic of proofs with strong provability operator (1994)] the logics of proofs BGrz, FGrz and MGrz were introduced that extend the logic Grz [see \textit{G. Boolos}, The logic of provability (1993; Zbl 0891.03004)]. The author shows that the logic BGrz possesses the strong Craig property while FGrz and MGrz do not possess even the weak Craig property.