Two-point convex-hull conjecture for the secrecy-channel curve

Let $\boldsymbol{p}$ range over $[0,0.5]$, and define the 3D space curve

$$\mathscr{C}=\{(p,h(\epsilon+p-2\epsilon p),h(\delta+p-2\delta p))\mid p\in [0,0.5]\}.$$

Two-point convex-hull conjecture. Any point in the convex hull of $\mathscr{C}$ can be expressed as the convex combination of only two points belonging to $\mathscr{C}$. This geometric claim would imply that only two values of the parameter controlled by the auxiliary random variable are needed, and hence that the auxiliary random variable can be taken to be binary; the supplied text presents it as an unresolved conjecture supporting the binary-alphabet reduction.