Can every integer N≥2 be written as a ratio of two products of consecutive integers? (Erdős #686)

Can every integer $N\geq 2$ be written as $$N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$ for some integers $k\geq 2$ and $m\geq n+k$ (with $n\geq 0$)? Here the numerator and denominator are each products of $k$ consecutive integers, and the condition $m\geq n+k$ forces the numerator's block $\{m+1,\ldots,m+k\}$ to lie strictly above the denominator's block $\{n+1,\ldots,n+k\}$. A secondary question: for fixed $n$ and $k$, what can be said about the set of integers representable in this form?

Acceptance. This is an OPEN, proof-shaped universal statement. FULLY RESOLVES: a complete proof — machine-checkable (Lean/Coq; a formal statement already exists in the formal-conjectures repo) preferred, otherwise a full written proof — that every integer $N\geq 2$ admits such a representation (e.g. an explicit construction producing valid $(k,m,n)$ for each $N$ together with a proof of validity), OR a proof that some specific integer $N\geq 2$ has no representation of this form (a single such $N$ with a proof of non-representability resolves the question negatively). ADVANCES: prove representability for an infinite family of $N$ (for example all $N$ in a fixed residue class, all sufficiently large $N$, or all $N$ of a stated form) with proof; or establish structural constraints on the set of representable integers; or, for fixed $n$ and $k$, characterise the representable $N$. A finite computation exhibiting representations for all $N$ up to some bound is a supporting certificate but does not settle the universal claim. Deliver the proof, or the construction together with its validity argument.