The candidate function conjecture for the finite-dimensional functional inequality

Let $\mathcal{S}$ be the set of pairs of random variables $(U,W)\in(0,1)^2$ with prescribed values of $\mathbb{E}[U]$, $\mathbb{E}[W]$, $\mathbb{E}[H_2(U)]$, and $\mathbb{E}[H_2(W)]$. Define

$$\zeta(m_u,m_w,e_u,e_w)=\inf_{(U,W)\in\mathcal{S}}\frac12\mathbb{E}[(U-W)(J(W)-J(U))].$$

Let $\Psi$ be the class of functions satisfying the functional inequality defining the paper's finite-dimensional formulation, and define $\eta$ and $\phi$ by

$$\eta(x)=(1-2H_2^{-1}(x))J(H_2^{-1}(x)),$$

with $r$ determined as in the paper, and

$$\phi(x,y)=\begin{cases}\eta(y)-\dfrac{y}{r}\eta(r),&H_2(x)>y,\\0,&H_2(x)\leq y.\end{cases}$$

The candidate function conjecture. The function $\phi$ belongs to the class $\Psi$. If true, this would establish the most-informative Boolean function conjecture for balanced functions. The claim is a finite-dimensional inequality supported by numerical evidence; its general proof remains open.