The conjecture that comodule equivalence is generated by weak equivalences

Let $(A,\Gamma)$ and $(B,\Sigma)$ be flat Hopf algebroids, and let a weak equivalence mean a morphism of flat Hopf algebroids inducing an equivalence between their comodule categories. The conjecture on Hopf algebroid comodule equivalences. If the category of $\Gamma$-comodules is equivalent to the category of $\Sigma$-comodules, then $(A,\Gamma)$ and $(B,\Sigma)$ are connected by a chain of weak equivalences. This is a Morita-type question for fundamentally commutative Hopf algebroids: unlike general rings, the authors do not expect nontrivial Morita phenomena. The statement is presented as a conjecture and no resolution is supplied in the source.