Frantzikinakis–Lesigne–Weirdl conjecture on random differences in Szemerédi's theorem

Let $k$ be a positive integer and let $S\subset\mathbb{N}$ be chosen independently at random with

$$\mathbb{P}[d\in S]=\omega\left(\frac{1}{d}\right).$$

Here, $f=o(g)$ if and only if $g=\omega(f)$. Frantzikinakis–Lesigne–Weirdl conjecture. Asymptotically almost surely, every subset of $\mathbb{N}$ with positive upper density contains a $k$-term arithmetic progression whose common difference belongs to $S$. This conjecture asks for the threshold density of a random set of allowed common differences ensuring Szemerédi's theorem; the source presents it as an open conjecture concerning random restrictions of arithmetic-progression differences.