The subspace approximation conjecture for functions on $\mathbb{F}_2^n$

Let $G=\mathbb{F}_2^n$, let $f\in A(G)$, and let $\epsilon\in(0,1]$. Write

$$A_f:=\|f\|_{A(G)}\|f\|_{L^\infty(G)}^{-1}.$$

Here $\operatorname{codim}V$ denotes the codimension of a subspace $V\leq G$, and $\|\cdot\|_{L^2(x'+V)}$ is the normalized $L^2$-norm on the coset $x'+V$. Subspace approximation conjecture. There is a subspace $V$ of $G$ such that

$$\operatorname{codim}V\ll\epsilon^{-2}A_f,$$

and

$$\sup_{x'\in G}\|f-f(x')\|_{L^2(x'+V)}\leq\epsilon\|f\|_{L^\infty(G)}.$$

The statement is proposed as a likely optimal bound for the model analogue of the paper’s main theorem. The source does not report a proof or disproof, so its status remains open.