If $R$ and $S$ are Morita equivalent rings, then the monoidal categories of $R$-bimodules and $S$-bimodules are monoidally equivalent.
Now, consider the converse:
Question: If the monoidal categories of $R$-bimodules and $S$-bimodules are monoidally equivalent, must $R$ and $S$ be Morita equivalent?
In Equivalent bimodule categories, it was asked whether the same is true if we merely assume that the categories of bimodules are equivalent as plain categories. The above question is about the stronger hypothesis.