Conjecture $f\cap^{\mathsf{FCD}} g = f\cap g$ for every binary relations $f$ and $g$.
The counter-example is $f = {(=)}|_{\mho}$ and $g = \mho\times\mho \setminus f$. I proved $f \cap^{\mathsf{FCD}} g = {(=)} |_{\Omega}$ (where $\Omega$ is the Frechet filter object).