Kirshner–Samorodnitsky conjecture on additive energy on Hamming spheres

For $n$ and $k$, let $S(n,k)\subseteq\mathbb{F}_2^n$ be the set of vectors with exactly $k$ ones, let $A=S(n,k)$, and let $N=2^n$. For a function $f:\mathbb{F}_2^n\to\mathbb{R}$, define

$$\lVert f\rVert_{u_2}^4=\mathop{\mathbb{E}}_{\substack{a_1,a_2,a_3,a_4\in\mathbb{F}_2^n\a_1+a_2=a_3+a_4}}f(a_1)f(a_2)f(a_3)f(a_4),$$ $$\lVert f\rVert_2^2=\mathop{\mathbb{E}}_{a\in\mathbb{F}_2^n}f(a)^2,$$

and

$$\mu(A)=\max_{\substack{f:\mathbb{F}_2^n\to\mathbb{R}\operatorname{supp}(f)\subseteq A}}\frac{\lVert f\rVert_{u_2}^4}{\lVert f\rVert_2^4}.$$

Kirshner–Samorodnitsky conjecture. One has

$$\mu(A)=\frac{1}{N}\frac{E(A)}{|A|^2}.$$

Equivalently, the ratio $\lVert f\rVert_{u_2}^4/\lVert f\rVert_2^4$ is maximized when $f$ is constant. The conjecture concerns extremizers for additive energy among functions supported on a Hamming sphere; the supplied paper's abstract says that it proves this assertion, so the conjecture is treated here as solved.