Is there a composite $n+k$ with least prime factor $p(n+k)>k^2$ for all large $n$? (Erdős #681)

Let $p(m)$ denote the least prime factor of $m$. Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and $$p(n+k)>k^2?$$ That is, near every large $n$ one can find an index $k$ for which the shifted integer $n+k$ is composite yet has least prime factor exceeding $k^2$ (the compositeness requirement rules out the trivial case where $n+k$ is itself prime and $p(n+k)=n+k$).

Acceptance. FULLY RESOLVES: a complete proof — a Lean/Coq formalisation matching the DeepMind formal-conjectures statement for #681 is preferred, otherwise a full written proof — that for all large $n$ there is a $k$ with $n+k$ composite and $p(n+k)>k^2$, or a proof that this fails. ADVANCES: prove the statement under a precisely stated prime-distribution hypothesis; or prove the generalisation with $k^2$ replaced by $k^d$ for a specific $d>1$ (conditional or unconditional, clearly flagged); or computationally verify that every $n$ up to a record $N$ admits such a composite $n+k$, supplying the search program and the attained $N$ strictly beyond any bound recorded here, and/or extend OEIS A389680. Deliver the proof or the verification code with its attained $N$.