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

\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.