  Motivated by applications to matrix multiplication algorithms, Pratt asked
(ITCS'24) how large a subset of $[n] \times [n]$ could be without containing a
skew-corner: three points $(x,y), (x,y+h),(x+h,y')$ with $h \ne 0$. We prove
any skew corner-free set has size at most $\exp(-\Omega(\log^{1/12} n))\cdot
n^2$, nearly matching the best known lower bound of $\exp(-O(\sqrt{\log
n}))\cdot n^2$ by Beker (arXiv'24). Our techniques generalize those of Kelley
and Meka's recent breakthrough on three-term arithmetic progression (FOCS'23),
answering a question of Beker (arXiv'24). We note that a similar bound was
obtained concurrently and independently by Mili\'cevi\'c (arXiv'24).
