Earlier I claimed that I proved the following theorem:
$latex (\mathcal{A}\ltimes\mathcal{B})\sqcap(\mathcal{A}\rtimes\mathcal{B})=\mathcal{A}\times_{F}^{\mathsf{RLD}}\mathcal{B}$ for every filters $latex \mathcal{A}$, $latex \mathcal{B}$ on sets.
(Here $latex \ltimes$ and $latex \rtimes$ 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.