This question is a background for my previous question.

Suppose $A$ and $B$ are two algebras over $\mathbb{C}$ with the sequences of norms $\lbrace\|\cdot\|_{\Xi,n}\rbrace$ and on $M_n(\Xi)$, $\Xi\in\lbrace A, B\rbrace$, satisfying the conditions of Blecher-Ruan-Sinclar theorem (so that, if I understand it right, we may construct concrete representations). Suppose also that $f\colon A \to B$ is a completely bounded map that has a completely bounded inverse $f^{-1}\colon B\to A$.

Can we somehow establish an isomorphism between categories of rigged modules over $A$ and $B$. And if yes, is there any good reference?

It can probably fit into the notion of (P)-context, but I can't reach the book right now to check all the conditions.