Intersection with a staroid product through its upper sets

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

Conjecture. Let $𝔉$ is a family of sets of filters on distributive lattices with least elements. Let $a\in \prod 𝔉$, $S\in 𝒫\prod 𝔉$ is a generalized filter base, $\bigsqcup S=a$, $f$ is a staroid of the form $\prod 𝔉$. Then

$$\prod ^{\mathrm{Strd}(𝔉)}a\not\asymp f\iff \forall A\in S:\prod ^{\mathrm{Strd}(𝔄)}A\not\asymp f.$$

(This conjecture may be weakened for the special case of filters on powersets.)