A surprisingly hard problem

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.

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