Estimate $S(k)$, the least $x$ forcing dense $k$-runs each divisible by a prime $\leq x$ (Erdős #929)

For a large integer $k\geq 2$, let $S(k)$ be the least $x$ for which there is a set of positive (natural) density of integers $n$ such that every one of $$n+1,\ n+2,\ \ldots,\ n+k$$ is divisible by some prime $\leq x$. Estimate $S(k)$; in particular, is it true that $$S(k)\geq k^{1-o(1)}?$$

Acceptance. FULLY RESOLVES (proof-shaped): a complete written or machine-checkable proof determining the order of $S(k)$ — in particular proving (or disproving) the lower bound $S(k)\geq k^{1-o(1)}$, ideally with matching upper and lower bounds pinning the growth rate. ADVANCES (checkable milestones): improve the lower bound strictly beyond the recorded $S(k)>k^{1/2-o(1)}$ toward $k^{1-o(1)}$, with proof; improve the upper bound below the stated Ford–Green–Konyagin–Maynard–Tao bound $S(k)\ll k\,\log\log\log k/(\log\log k\,\log\log\log\log k)$, with proof; or a certified computation, for a documented range of small $k$, of the least $x$ for which dense runs of $k$ consecutive integers each with a prime factor $\leq x$ appear, with a reproducible program and the resulting data. Deliver the proof/formalisation or the reproducible computation with certified output.