I have found a surprisingly easy proof of this conjecture which I proposed yesterday.

Theorem Let $S$ be a set of binary relations. If for every $X, Y \in S$ we have $\mathrm{up} (X \sqcap^{\mathsf{FCD}} Y) \subseteq S$ then there exists a funcoid $f$ such that $S = \mathrm{up}\, f$.

The proof (currently available in this PDF file) is based on “Funcoids are filters” chapter of my book.