The following conjecture seems trivial but I have a hard hour trying to prove it. I suspect I have a big difficulty on the course of my research.
Conjecture $latex \prod^{\mathsf{FCD}} a \not\asymp\prod^{\mathsf{FCD}} b \Leftrightarrow \forall i\in n : a_i \not\asymp b_i$ for every $latex n$-indexed (where $latex n$ is an arbitrary index set) families of filters $latex a$ and $latex b$.
See This Web page and specifically this draft article for the notation I use.
It seems I have found a proof. See a proof sketch (with minor errors) in this rough draft.
Well, there is an unfounded step in this proof. I keep trying to prove it.
I give up. I am going to publish it as a conjecture.