Shur's asymptotic growth conjecture for power-free languages

For any real $b2>1$, let $b1(k,b2)$ denote the growth rate of the language of $b2$-free words over an alphabet of size $k$. Shur's asymptotic growth conjecture. For every fixed integer $ngeq3$ and arbitrarily large integer $k$,

$$\alpha\left(k,\frac{n}{n-1}\right)=k+1-n-\frac{n-1}{k}+O\left(\frac{1}{k^2}\right)$$

and

$$\alpha\left(k,\frac{n}{n-1}^+\right)=k+2-n-\frac{n-1}{k}+O\left(\frac{1}{k^2}\right).$$

The paper proves the asymptotic lower bound and states that this establishes the conjecture for $\frac{9}{8}\leq\beta<2$, while the general case with $\beta<2$ remains open.