Estimate $f(k,n)$: primes needed to over-cover a $k$-subset of $\{1,\ldots,n\}$ (Erdős #983)

Let $n\ge 2$ and $\pi(n)<k\le n$, where $\pi$ is the prime-counting function. Define $f(k,n)$ to be the smallest integer $r$ such that: for every $A\subseteq\{1,\ldots,n\}$ with $\lvert A\rvert=k$, there exist primes $p_1,\ldots,p_r$ for which more than $r$ elements $a\in A$ have all of their prime factors among $\{p_1,\ldots,p_r\}$ (i.e. $a$ is $\{p_1,\ldots,p_r\}$-smooth). Estimate $f(k,n)$, particularly in the range $\pi(n)+1<k=o(n)$. In particular, is it true that $$2\pi(n^{1/2})-f(\pi(n)+1,n)\to\infty$$ as $n\to\infty$?

Acceptance. FULLY RESOLVES: a proof (machine-checkable preferred, else a complete written proof) that $2\pi(n^{1/2})-f(\pi(n)+1,n)\to\infty$ (or that it stays bounded), together with a determination of the order of magnitude of $f(k,n)$ throughout the range $\pi(n)+1<k=o(n)$. ADVANCES (proof or reproducible certificate required): (a) prove the specific limit $2\pi(n^{1/2})-f(\pi(n)+1,n)\to\infty$, or a lower bound on this difference tending to infinity; (b) sharpen the Erdős–Straus asymptotic for $f(\pi(n)+1,n)$ or for $f(cn,n)$ with an improved error term; (c) obtain a new asymptotic or two-sided bound for $f(k,n)$ in an intermediate range $\pi(n)+1<k=o(n)$; or (d) compute exact values of $f(k,n)$ for small $n$ via a reproducible program with a certificate of the extremal subset. Deliver the proof/formalization, or the computation code plus certified values.