Is the reciprocal sum of running LCMs of $P$-smooth numbers irrational? (Erdős #269)

Let $P$ be a finite set of primes with $|P|\ge 2$, and let $a_1<a_2<\cdots$ enumerate the $P$-smooth positive integers, i.e. $\{a_1<a_2<\cdots\}=\{n\in\mathbb{N}: \text{if } p\mid n \text{ then } p\in P\}$. Writing $[a_1,\ldots,a_n]$ for the least common multiple of the first $n$ terms, is $$\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]}$$ irrational?

Acceptance. FULLY RESOLVES: prove that for every finite set of primes $P$ with $|P|\ge 2$ the sum $\sum_n 1/[a_1,\ldots,a_n]$ is irrational, OR exhibit a specific finite $P$ for which the sum is rational, with the rational value certified. A machine-checkable Lean 4 proof extending ErdosProblems/269.lean is the gold standard; otherwise a complete written proof with every step. ADVANCES: settle a genuine special case of the stated (with-repeats) problem beyond the easy infinite-$P$ case — e.g. a full proof for $|P|=2$ (all two-prime sets, or a specific pair such as $P=\{2,3\}$), or an extension of Erdős's duplicate-removed argument that rigorously controls the contribution of the repeated summands; each with proof. Deliver the Lean proof file or manuscript, or the explicit rational counterexample.