Conjecture on convexity and concavity of the conditional Rényi information-combining bound

Let $h_\alpha$ denote the binary conditional Rényi-entropy function and let $\ast$ denote binary convolution. There exists a value $1<\hat\alpha<1.3863$ such that

$$h_{\alpha}\left({h_{\alpha}}^{-1}(x)\ast {h_{\alpha}}^{-1}(y)\right)$$

is convex for $0<\alpha<\hat\alpha$ and concave for $\alpha\geq2$. This conjecture is based on numerical evidence following counterexamples to convexity and concavity in an intermediate range; the precise threshold below $1.3863$ remains open.