Is there a slowly growing 'good' pairwise-coprime sieving sequence? (Erdős #1101)

Let $u=\{u_1<u_2<\cdots\}$ be a sequence of integers that are pairwise coprime ($(u_i,u_j)=1$ for all $i\neq j$) and satisfy $\sum_i \frac{1}{u_i}<\infty$. Let $\{a_1<a_2<\cdots\}$ be the sequence of integers not divisible by any $u_i$. For $x$ define $t_x$ by $$u_1\cdots u_{t_x}\leq x< u_1\cdots u_{t_x}u_{t_x+1}.$$ Call the sequence $u$ good if for every $\epsilon>0$, for all sufficiently large $x$, $$\max_{a_k<x}(a_{k+1}-a_k) < (1+\epsilon)\,t_x\prod_i\left(1-\frac{1}{u_i}\right)^{-1}.$$ Is there a good sequence with $u_n<n^{O(1)}$ (polynomial growth)? Is there a good sequence with $u_n\leq e^{o(n)}$ (subexponential growth)?

Acceptance. FULLY RESOLVES: a complete proof exhibiting a good sequence with $u_n<n^{O(1)}$ or proving none exists; and/or a complete proof exhibiting a good sequence with $u_n\leq e^{o(n)}$ or proving none exists — machine-checkable (Lean) preferred, else a full rigorous written proof (with the goodness of any constructed sequence proved, not merely tested). ADVANCES (each independently checkable): a proof that some good sequence exists with growth strictly slower than any benchmark construction stated in the background (state the benchmark growth rate in words and beat it); a resolution of the strong form of #208, deciding whether the prime squares $u_i=p_i^2$ are good; or a new proved bound on the maximal-gap function for an explicit candidate sequence. Deliver the proof or construction together with its verification.