A proof of the following conjecture (now a theorem) was quickly found by me after its formulation:
Theorem $latex \left\langle f \right\rangle \mathcal{X} = \bigcap^{\mathfrak{F}} \left\{ \left\langle F \right\rangle \mathcal{X} | F \in \mathrm{up}f \right\}$ for every funcoid $latex f$ and f.o. $latex \mathcal{X}$.
See the updated version of my article “Funcoids and Reloids” for the proof.