Bound the powerful part $Q_2$ of a product of consecutive integers (Erdős #935)

For an integer $n=\prod_p p^{k_p}$, let $Q_2(n)$ be its powerful part, retaining only the prime powers whose exponent is at least $2$: $$Q_2(n) = \prod_{p:\, k_p\geq 2} p^{k_p}.$$ Three questions are asked about the powerful part of a block of consecutive integers. (1) Is it true that for every $\epsilon>0$ and every $\ell\geq 1$, if $n$ is sufficiently large then $$Q_2\bigl(n(n+1)\cdots(n+\ell)\bigr) < n^{2+\epsilon}\,?$$ (2) For $\ell\geq 2$, is $$\limsup_{n\to\infty}\frac{Q_2\bigl(n(n+1)\cdots(n+\ell)\bigr)}{n^2}=\infty\,?$$ (3) For $\ell\geq 2$, is $$\lim_{n\to\infty}\frac{Q_2\bigl(n(n+1)\cdots(n+\ell)\bigr)}{n^{\ell+1}}=0\,?$$ One may pose all of these with $Q_2$ replaced by $Q_r$ for any $r>2$, keeping only the prime powers of exponent $\geq r$.

Acceptance. FULLY RESOLVES: a complete proof (Lean/Coq machine-checkable preferred, else a full rigorous written proof) settling question (1) — the bound $Q_2(n(n+1)\cdots(n+\ell))<n^{2+\epsilon}$ for all $\ell\geq 1$ — and/or question (3) — the limit $Q_2(n(n+1)\cdots(n+\ell))/n^{\ell+1}\to 0$ for $\ell\geq 2$ — for $Q_2$ or general $Q_r$; a proof may be abc-conditional if that dependence is clearly flagged. Question (2) is already resolved affirmatively (van Doorn's Pell construction), so a resolution must address (1) or (3). ADVANCES (each independently checkable): establish a non-trivial unconditional upper bound toward (1) strictly improving on any bound stated in the background; produce, with reproducible code, record-large values of $Q_2(n(n+1)(n+2))/n^2$ extending the known construction and documenting the search range; or prove a conditional partial result under a clearly stated hypothesis. Deliver the proof or the search program plus the attained data and certificates.