Shur and Gorbunova's asymptotic growth-rate conjecture for threshold languages

For every $n\geq 2$, let $T_n$ be the threshold language of order $n$, and define its growth rate by

$$\alpha(T_n)=\limsup_{k\rightarrow\infty}(C_{T_n}(k))^{1/k}.$$

Since $T_n$ is factorial, this limsup can equivalently be replaced by a limit. Shur and Gorbunova's conjecture. The sequence of growth rates $\{\alpha(T_n)\}$ converges, as $n$ tends to infinity, to a limit $\hat{\alpha}\approx 1.242$. The conjecture concerns the asymptotic behaviour of the threshold-language growth rates; the source provides lower bounds for $\alpha(T_n)$ but does not establish this limiting value.