The residue-class formulation of the divisor-set conjecture

Let $n$ be a positive integer, and let $\mathbb{Z}^{\widetilde{O}(n^{n^{1/3}})}$ denote the integers in the indicated size range. The residue-class formulation. There exists a set $S\subseteq\mathbb{Z}^{\widetilde{O}(n^{n^{1/3}})}$ with $k=|S|=\widetilde{O}(n^{1/3})$, and a set $T$ with $|T|<k$, such that for every $j\in\{1,\ldots,n\}$ and every $s\in S$, the residue $s\bmod j$ belongs to $T$.

This is another general formulation of the desired number-theoretic structure. The paper states that the more general conjectures are not presently known to yield algorithms, and gives no evidence of resolution.