I earlier proclaimed that I positively solved this conjecture:
Conjecture $latex f\cap^{\mathsf{FCD}} g = f\cap g$ for every binary relations $latex f$ and $latex g$.
There were error in my proof and I deleted it. See the draft of Funcoids and Reloids article for details on the current state of the problem.
A counter-example for this conjecture appears in the appendix “Some counter-examples” of the last version of Funcoids and Reloids article.