$J$-equivalence for associative algebras

We study the combinatorics of an analogue of Green's $\mathcal{J}$-relation (a.k.a. the two-sided relation) for the bicategory of finite-dimensional bimodules over finite-dimensional associative algebras over a fixed field. In particular, we provide a number of exameples for both $\mathcal{J}$-equivalent and inequivalent but $\mathcal{J}$-comparabe algebras.