The polynomial Freiman-Ruzsa conjecture states that there exists $k > 0$, such that for all $\epsilon > 0$, and for all large $n$ and all functions $ f: \mathbb{F}_2^n \mapsto\mathbb{F}_2^n$, the following holds. If

$$Pr_{x, x' \in \mathbb{F}_2^n} [f(x) + f(x') = f(x+x')] \geq \epsilon \;,$$ then there is a matrix $M \in \mathbb{F}_2^{n \times n}$ such that

$$Pr_{x \in \mathbb{F}_2^n} [ f(x) = M x] \geq \epsilon^k]\;.$$

My question is whether we can prove this conjecture for the special case when $f$ is known to be a bijection.