Are there only finitely many pairs of integer intervals whose reciprocal sums total an integer? (Erdős #288)

For a finite interval $I\subset\mathbb{N}$ (a set of consecutive integers), write $\sum_{n\in I}\frac1n$ for the sum of the reciprocals of its elements. Is it true that there are only finitely many pairs of intervals $I_1,I_2$ with $$\sum_{n\in I_1}\frac1n+\sum_{n\in I_2}\frac1n\in\mathbb{N}?$$ The question is open even in the restricted case $\lvert I_2\rvert=1$ (a single reciprocal), and one may ask more generally whether, for each fixed $k$, only finitely many $k$-tuples of intervals have reciprocal sums totalling an integer.

Acceptance. FULLY RESOLVES (proof-shaped): a complete proof that only finitely many pairs $(I_1,I_2)$ satisfy $\sum_{I_1}1/n+\sum_{I_2}1/n\in\mathbb{N}$ (ideally with an effective bound on the endpoints), or a proof that infinitely many exist (e.g. an explicit infinite family). A machine-checkable proof is preferred, otherwise a full written proof; a computation alone cannot establish finiteness. ADVANCES (each independently checkable): (a) an exhaustive, certified enumeration of ALL such pairs with endpoints up to a stated bound $M$ (reproducible program plus an exhaustiveness certificate), extending the range over which the complete solution list is known; (b) a proof of finiteness under a restriction (e.g. $\lvert I_2\rvert=1$, or a bounded target integer value); (c) a rigorous upper bound on the number or the size of solutions. Deliver the proof, or the enumeration code plus the certified list and bound.