Green's removal-property conjecture for homogeneous linear systems

Let $M$ be an $\ell\times p$ integer matrix of rank $\ell$, and consider the homogeneous system of linear equations $Mx=0$. A set $S\subseteq[n]$ is $(M,0)$-free if it contains no vector $x\in S^p$ satisfying $Mx=0$. The system has the removal property if, for every $\delta>0$, there is an $\epsilon=\epsilon(\delta,M,0)>0$ such that whenever $S\subseteq[n]$ contains at most $\epsilon n^{p-\ell}$ solutions in $S^p$ to $Mx=0$, at most $\delta n$ elements can be removed from $S$ to obtain an $(M,0)$-free set.

Green's conjecture. Every system of homogeneous linear equations $Mx=0$ has the removal property.

Green proved this for a single homogeneous linear equation. The conjecture asks for the corresponding removal statement for arbitrary systems and is resolved by the main result of the paper using the hypergraph removal lemma.