Quasipolynomial bounds for the corners theorem

Let $G$ be a finite abelian group and $A$ be a subset of $G \times G$ which is corner--free, meaning that there are no $x, y \in G$ and $d \in G \setminus \{0\}$ such that $(x, y)$, $(x+d, y)$, $(x, y+d) \in A$. We prove that $$|A| \le |G|^2 \cdot \exp(-(\log |G|)^{\Omega(1)}).$$ As a consequence, we obtain polynomial (in the input length) lower bounds on the nondeterministic communication complexity of Exactly-N in the 3-player Number-on-Forehead model. We also obtain the first "reasonable'' lower bounds on the coloring version of the $3$-dimensional corners problem, as well as on the nondeterministic communication complexity of Exactly-N in the 4-player Number-on-Forehead model.