Conjecture on convexity and concavity of the Arimoto information-combining bound

Let $k_\alpha^A$ denote the Arimoto conditional-entropy function and let $\ast$ denote binary convolution. There exists a value $1<\hat\alpha<1.5783$ such that

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

is convex for $0<\alpha<1$ and concave for $1<\alpha\leq\hat\alpha$. This conjecture is based on numerical evidence; the stated threshold is bounded by verified counterexamples and additional numerical observations, while the behavior beyond $\hat\alpha$ is not asserted.