Almost-homomorphism formulation of the Polynomial Freiman-Ruzsa conjecture

Let $f:\mathbb{F}_2^n\rightarrow\mathbb{F}_2^{n'}$ be a map, and let $S$ be a set of size $K$ such that

$$f(x+y)-f(x)-f(y)\in S$$

for all $x,y\in\mathbb{F}_2^n$.

Almost-homomorphism formulation of the Polynomial Freiman-Ruzsa conjecture. Is it true that $f=\widetilde f+g$, where $\widetilde f:\mathbb{F}_2^n\rightarrow\mathbb{F}_2^{n'}$ is linear and $|\operatorname{im}(g)|\leqslant K^C$?

This is an equivalent formulation of the Polynomial Freiman-Ruzsa conjecture. The bound $K^C$ can be replaced trivially by $2^K$, while Sanders's result gives the weaker bound $\exp(\log^{4+o(1)}K)$.