Finitely many pairs of consecutive-integer blocks (lengths ≥3) with identical prime support? (Erdős #931)

Let $k_1\geq k_2\geq 3$ be fixed integers. Are there only finitely many pairs $(n_1,n_2)$ with $n_2\geq n_1+k_1$ (and $n_1\geq 0$) such that the two products $$\prod_{1\leq i\leq k_1}(n_1+i)\quad\text{and}\quad\prod_{1\leq j\leq k_2}(n_2+j)$$ have the same set of prime factors? Equivalently, are there only finitely many pairs of blocks — one of $k_1$ and one of $k_2$ consecutive integers, the second lying above the first — whose products share identical prime support (the same radical)?

Acceptance. This is an OPEN, proof-shaped finiteness problem. FULLY RESOLVES: for the stated regime $k_1\geq k_2\geq 3$, a complete proof — machine-checkable (Lean/Coq; a formal statement already exists in the formal-conjectures repo) preferred, otherwise a full written proof — that only finitely many pairs $(n_1,n_2)$ with $n_2\geq n_1+k_1$ give products of identical prime support, OR a proof that infinitely many such pairs exist (for example an explicit infinite family, with proof). ADVANCES: settle a genuine sub-case with proof — e.g. prove finiteness (or infinitude) for a specific small pair such as $k_1=k_2=3$ or $k_1=k_2=4$; or prove finiteness conditional on a clearly stated hypothesis (such as the abc conjecture); or exhaustively search all blocks up to a new record bound and deliver a reproducible certificate of the complete list of matching-support pairs found (extending Tijdeman's and AlphaProof's examples). A search alone cannot establish finiteness. Deliver the proof (formal, written, or clearly-flagged conditional), or the search program together with its exhaustiveness certificate and catalogued examples.