Like a complete idiot, this took me a few years to disprove my conjecture, despite the proof is quite trivial.

Here is the complete solution:

Example $latex [S]\ne\{\bigsqcup^{\mathfrak{A}}X \mid X\in\mathscr{P} S\}$, where $latex [S]$ is the complete lattice generated by a strong partition $latex S$ of filter on a set.

Proof Consider any infinite set $latex U$ and its strong partition $latex \{\uparrow^U\{x\} \mid x\in U\}$.

$latex \{\bigsqcup^{\mathfrak{A}}X \mid X\in\mathscr{P} S\}$ consists only of principal filters.

But $latex [S]$ obviously contains some nonprincipal filters.

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