I’ve proved:

$latex \bigsqcap \langle \mathcal{A} \times^{\mathsf{RLD}} \rangle T =

\mathcal{A} \times^{\mathsf{RLD}} \bigsqcap T$ if $latex \mathcal{A}$ is a filter and $latex T$ is a set of filters with common base.

$latex \bigsqcup \left\{ \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}

\hspace{1em} | \hspace{1em} \mathcal{B} \in T \right\} \neq \mathcal{A}

\times^{\mathsf{RLD}} \bigsqcup T$ for some filter $latex T$ and set of filters $latex T$ (with a common base).

See preprint of my book.