Intersecting two staroidal products

Prerequisites: http://www.mathematics21.org/algebraic-general-topology.htmlAlgebraic General Topology.

Conjecture. ${\prod }^{\mathrm{Strd}}a\not\asymp {\prod }^{\mathrm{Strd}}b\iff b\in {\prod }^{\mathrm{Strd}}a\iff a\in {\prod }^{\mathrm{Strd}}b\iff \forall i\in n:a_i\not\asymp b_i$ for every $n$-indexed families $a$ and $b$ of filters on powersets.

The above conjecture has the below consequence (see my book for not so long proof).

Conjecture. Let $f$ is a staroid on powersets and $a\in {\prod }_{i\in \mathrm{arity}f}\mathrm{Src}{f}_i$, $b\in {\prod }_{i\in \mathrm{arity}f}\mathrm{Dst}{f}_i$. Then

$$\prod ^{\mathrm{Strd}}a\left[\prod ^{(C)}f\right]\prod ^{\mathrm{Strd}}b\iff \forall i\in n:a_i[f_i]b_i.$$