The skew corner-free set bound

Let $[n]=\{1,\ldots,n\}$, and let $S\subseteq[n]^2$ be skew corner-free, meaning that $S$ contains no three points of the form

$$(x,y),\ (x,y+\delta),\ (x+\delta,y')$$

with $\delta\ne0$. The skew corner-free set conjecture. For every $\varepsilon>0$, one has

$$|S|\le O(n^{1+\varepsilon}).$$

The trivial construction gives $|S|\ge n$, while Petrov's construction gives $|S|\ge\Omega(n\log n/\sqrt{\log\log n})$. The best upper bound stated here is only $O(n^2/(\log\log n)^{0.0137\cdots})$, so the conjectured near-linear bound remains open.