The kappa inequalities for the finite-dimensional functional approach

For $(u,w)\in(0,1)^2$, define

$$\kappa(u,w)=\frac{(u-w)(J(w)-J(u))}{2}-|u-w|J\left(L^{-1}\left(\frac{H_2(u)+H_2(w)}{|u-w|}\right)\right),$$

with $\kappa(u,u)=0$. The kappa inequalities. The following two assertions hold: for $u,w\in(0,1/2]$, $\kappa(u,w)\leq\kappa(1-u,w)$; and the function $(u,w)\mapsto\kappa(H_2^{-1}(u),H_2^{-1}(w))$ is jointly convex. These inequalities would yield the lower bound needed to prove that $\phi\in\Psi$, and hence would imply the balanced case of the most-informative Boolean function conjecture. They are supported by numerical evidence but are not proved in the paper.