OPEN

Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let $\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\}$. Is the sum
\[\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]},\]
where $[a_1,\ldots,a_n]$ is the lowest common multiple of $a_1,\ldots,a_n$, rational or irrational?

If $P$ is infinite this sum is always irrational.