The explicit lower-bound conjecture for zeta

Let $m_u,m_w,e_u,e_w\in[0,1]$ satisfy $H_2(m_u)\geq e_u$ and $H_2(m_w)\geq e_w$. Let $\phi$ and $J$ be the functions defined in the paper. The explicit lower-bound conjecture. One has

$$\begin{aligned} &\phi\left(\frac{1-|H_2^{-1}(e_u)-H_2^{-1}(e_w)|}{2},\frac{e_u+e_w}{2}\right)\\ &+(H_2^{-1}(e_u)-H_2^{-1}(e_w))\frac{J(H_2^{-1}(e_w))-J(H_2^{-1}(e_u))}{2}\\ &-\phi\left(\frac{1-|m_u-m_w|}{2},\frac{e_u+e_w}{2}\right)\\ &\geq \phi\left(\frac{m_u+m_w}{2},\frac{e_u+e_w}{2}\right)-\frac12\phi(m_u,e_u)-\frac12\phi(m_w,e_w). \end{aligned}$$

This inequality, together with the preceding theorem in the paper, would establish that $\phi\in\Psi$. It is an open finite-dimensional inequality proposed as part of the route to the balanced most-informative Boolean function conjecture.