Effective bounds for Roth's theorem with shifted square common difference

Let $S$ be a subset of $\{1,\dots ,N\}$ avoiding the nontrivial progressions $x,x+y^2-1,x+2(y^2-1)$. We prove that $\mathrm{\mid }S\mathrm{\mid }\ll N\mathrm{/}{\log }_{m}N$, where ${\log }_m$ is the $m$-fold iterated logarithm and $m\in \mathbf{N}$ is an absolute constant. This answers a question of Green.