Isomorphism of categories of rigged modules via completely bounded isomorphism of operator algebras

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 $M_n(Xi)$, $Xiinlbrace A, Brbrace$, satisfying the conditions of Blecher-Ruan-Sinclar theorem (so that, if I understand it right, we may construct concrete representations). Suppose also that $fcolon A to B$ is a completely bounded map that has a completely bounded inverse $f^{-1}colon Bto A$.

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.