Improvement of the $b8$-log-convexity threshold for MaxwellBoltzmann convolutions

Let $(\mu_n)$ be a $\theta$-log-convex Stieltjes moment sequence, and let $(\mu_n^{\circ t})$ denote its Maxwell-Boltzmann convolution semigroup for $t\in\mathbf{R}_+$. The preceding proposition gives a maximal admissible value of $\theta$ slightly less than $1/6$. Improvement conjecture. The estimate for this maximal $\theta$ can be improved by including further terms in the expansion of $\mu_n^{\circ t}$, beyond the first two terms. The stated bound uses only the first two terms of the expansion, so incorporating additional terms may yield a larger admissible threshold for $\theta$; the source does not give a specific improved value.