The Polynomial Freiman-Ruzsa conjecture for approximate homomorphisms

Let $V$ be a finite-dimensional vector space in characteristic $2$. Suppose that $f:V\rightarrow V$ satisfies

$$\{f(x+y)-f(x)-f(y):x,y\in V\}\subset S.$$

Polynomial Freiman-Ruzsa conjecture. There is a linear map $\tilde f:V\rightarrow V$ and a set $\tilde S$ with $|\tilde S|\ll |S|^C$ such that

$$\{f(x)-\tilde f(x):x\in V\}\subset \tilde S.$$

This is presented as an equivalent formulation of the Polynomial Freiman-Ruzsa conjecture, which asks for polynomial quantitative control in characteristic $2$. The surrounding discussion notes that the best bounds known in the cited work are of the form $e^{(\log K)^C}$, so the polynomial bound remains open.