A difficulty on the way of my research

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 \prod^{\mathsf{FCD}} a \not\asymp\prod^{\mathsf{FCD}} b \Leftrightarrow \forall i\in n : a_i \not\asymp b_i for every n-indexed (where n is an arbitrary index set) families of filters a and b.

See This Web page and specifically this draft article for the notation I use.

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