Approximate homomorphism testing conjecture

Let $f:\mathbb{F}_2^n \to \mathbb{F}_2^m$ be a function, and let $\epsilon$ measure the probability that it satisfies the additive homomorphism equation on uniformly sampled $x,y \in \mathbb{F}_2^n$. Approximate homomorphism testing conjecture. If

$$\Pr_{x,y}[f(x+y)=f(x)+f(y)] \ge \epsilon,$$

then there is a linear map $\ell:\mathbb{F}_2^n \to \mathbb{F}_2^m$ such that

$$\Pr_x[f(x)=\ell(x)] \ge \epsilon^{O(1)}.$$

This is an equivalent formulation of the Polynomial Freiman–Ruzsa conjecture in the source. The paper presents it as an open polynomial quantitative strengthening of approximate homomorphism testing.