The most general divisor-set conjecture

Let $n$ be a positive integer, and let divisibility of a product mean that each integer in $\{1,\ldots,n\}$ divides that product. The most general divisor-set conjecture. For every $n$, there exists a set $S$ of size $\widetilde{O}(n^{1/3})$, with each $s\in S$ of size $\widetilde{O}(n^{n^{1/3}})$, such that

$$1,\ldots,n\mid\prod_{s,t\in S}(s-t).$$

The paper presents this as a more general variation of the preceding structured conjectures, while noting that it is not currently known how to turn it into algorithms. No resolution evidence is supplied.