I am now trying to prove or disprove this innocently looking but somehow surprisingly hard conjecture:

Conjecture If $S$ is a generalized filter base then $\left\langle f \right\rangle \bigcap{\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} \left\langle\left\langle f \right\rangle \right\rangle S$ for every funcoid $f$.