Erroneous theorem turned into a conjecture

Earlier I claimed that I proved the following theorem:

(\mathcal{A}\ltimes\mathcal{B})\sqcap(\mathcal{A}\rtimes\mathcal{B})=\mathcal{A}\times_{F}^{\mathsf{RLD}}\mathcal{B} for every filters \mathcal{A}, \mathcal{B} on sets.

(Here \ltimes and \rtimes is what I call oblique products.)

Now I have found an error in my proof, so now it is presented as a conjecture in my book.

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