The supporting-functional inequality for the binary symmetric key-interaction conjecture

Let $\bar{a}:=1-a$ and $a*b:=a(1-b)+(1-a)b$. Let $\alpha,\epsilon,f,g\in(0,1)$, and define $s$ by

$$s:=\frac{(\bar{\epsilon}-\epsilon)[\log(\alpha*\epsilon)-\log(\bar{\alpha}*\epsilon)]}{\log\alpha-\log\bar{\alpha}}.$$

Define $c$ by the two equivalent expressions in the source, with the quantities $k$, $f*\bar{g}*\epsilon$, $h$, $P_{XY}$, and $(\hat X,\hat Y)$ interpreted as in the paper; when $\alpha=\frac12$, these expressions are defined by continuity.

Supporting-functional inequality.

$$sH(\hat X,\hat Y)-I(\hat X;\hat Y)\le s[h(\epsilon)+h(\alpha)]-[h(\alpha*\epsilon)-h(\epsilon)]+\frac{c(f-\frac12)(g-\frac12)}{f*\bar g*\epsilon}.$$

Equality holds at the four points specified in the source.

This is presented as a conjectured inequality that would imply the one-way tradeoff conjecture for binary symmetric sources. Because the supplied context does not define all notation needed to state it self-containedly, its precise formulation should be checked against the paper.