Do these set systems imply a partition? During research, I hit a set theoretic claim that I could neither proof nor disproof.

Let $S_1,S_2,S_3$ be three set systems over the same finite universe $U$ such that

$S_1,S_2,S_3$ are closed w.r.t. $\cap$ and $\cup$ and

for each pair $u,v \in U$, there are two disjoint sets $X \in S_i, Y \in S_j$, $i,j \in \{1,2,3\}$, $i \neq j$, such that $u \in X$ and $v \in Y$.


Then, there is a partition of $U$ into at most three sets, each from a different set system.
They don't necessarily imply a partition, here is a counterexample.
Each set in each $S_i$ will have an even size, while $U$ will have an odd number of elements, this already guarantees that a partition is impossible.
$U$ is the nine vertices of a cycle of length $9$, whose edges are colored with $3$ colors, in an alternating way.
With the union operation each color class generates a set system, these will be the $S_i$.
In more detail, the generating edges are 01, 34, 67 for $S_1$, 12, 45, 78 for $S_2$ and 23, 56, 80 for $S_3$.
It is easy to check that your separating condition holds.