Open. The only bound in hand is the trivial one, $q(n,\log n)\le(1+o(1))(\log n)^2$ for all $n$, Erdős's display (10) at $k=[\log n]$, which the site derives by comparing the primorial of $q(n,k)$ with the product: every prime below $q(n,k)$ divides $\prod_{1\le i\le k}(n+i)$, so their product is at most it. The construction accepted on Problem 457 gives $q(n,\log n)>A\log n$ for every fixed $A$ and infinitely many $n$. A sketch by Tao in that thread gives $q(n,\log n)>\frac{1-o(1)}2\frac{\log\log n}{\log\log\log n}\log n$ for infinitely many $n$; the site reports it as proved, but no accepted claim covers it, and the written version there, a claimed page, reaches only the $\gg$ form. Both are far below $(\log n)^2$, and a heuristic recorded by Tao in the thread of Problem 457 suggests $q(n,\log n)\ll\log n\log\log n/\log\log\log n$ for all $n$, which he regards as beyond current methods. No source improving the trivial bound was found in the search whose scope the Current assessment records; this is a bounded negative finding, not a certificate of openness.