I’ve proved the following conjecture:
Theorem Let $latex f$ be a staroid such that $latex (\mathrm{form}\, f)_i$ is an atomic lattice for
each $latex i \in \mathrm{arity}\, f$. We have
$latex \displaystyle L \in \mathrm{GR}\, f \Leftrightarrow \mathrm{GR}\, f \cap
\prod_{i \in \mathrm{dom}\, \mathfrak{A}} \mathrm{atoms}\, L_i \neq \emptyset $
\prod_{i \in \mathrm{dom}\, \mathfrak{A}} \mathrm{atoms}\, L_i \neq \emptyset $
for every $latex L \in \prod_{i \in \mathrm{arity}\, f} (\mathrm{form}\,
f)_i$ (where upgrading is taken on the primary filtrator).
The proof is based on transfinite recursion. See this online article for the proof.
The above proof was with an error. Now there is a counter-example in the same article.