Is $f(n,k)=(1-\rho(\alpha)+o(1))k$ for the count of $n+i$ with prime factor $>k$? (Erdős #1184)

For positive integers $n$ and $k$, let $P(m)$ denote the largest prime factor of $m$, and let $f(n,k)$ count the number of indices $1\leq i\leq k$ for which $P(n+i)>k$. Let $\rho$ be the Dickman function (the asymptotic density of smooth numbers, $\rho(\alpha)=\lim_{x\to\infty}\Psi(x,x^{1/\alpha})/x$). Is it true that whenever $\alpha>1$ satisfies $n=k^{\alpha+o(1)}$, one has $$f(n,k)=(1-\rho(\alpha)+o(1))\,k?$$

Acceptance. FULLY RESOLVES: a complete proof (machine-checkable preferred, else full written proof) that $f(n,k)=(1-\rho(\alpha)+o(1))k$ for all $\alpha>1$ with $n=k^{\alpha+o(1)}$. ADVANCES: prove the asymptotic for a new range of $\alpha$ — in particular any nontrivial two-sided bound in the range $\alpha\geq 2$, where the background records none — or strengthen Erdős's one-sided bounds (improve the constant $c_\alpha$ in the lower bound, or extend the upper bound $f(n,k)<(\alpha-1+o(1))k$ beyond $1<\alpha<2$). A reproducible large-scale computation certifying agreement of $f(n,k)/k$ with $1-\rho(\alpha)$ over a stated range of $(n,k)$ supports but does not by itself settle the conjecture. Deliver the proof, or the computation code plus the ranges covered and the measured $f(n,k)/k$ versus $1-\rho(\alpha)$.