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Conjecture ![$ L \in \mathrel{\left[ f \right]} \Rightarrow \mathrel{\left[ f \right]} \cap \prod_{i \in \operatorname{dom} \mathfrak{A}} \operatorname{atoms} L_i \neq \emptyset $](/files/tex/20328c795890b2a043f28afc705aecd5679f72d9.png)
for every multifuncoid 
of the form whose elements are atomic posets.
for every multifuncoid of the form whose elements are atomic posets. See Algebraic General Topology, especially the theory of multifuncoids for definitions of used concepts.
See this my online article and in this MathOverflow answer for a counter-example.
Bibliography
*Victor Porton. Algebraic General Topology
* indicates original appearance(s) of problem.
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