I’ve found the following counter-example, to this conjecture:

Example For a set $S$ of binary relations

$\forall X_0,\dots,X_n\in S:\mathrm{up}(X_0\sqcap^{\mathsf{FCD}}\dots\sqcap^{\mathsf{FCD}} X_n)\subseteq S$

does not imply that there exists funcoid $f$ such that $S=\mathrm{up}\, f$.

The proof is currently available at this PDF file and this wiki page.