Stronger tail-restricted growth conjecture for power-free languages

For a real $b2>1$, let $b1'(k,b2)$ be the growth rate of the language of words containing no $b2$-power whose tail has length at most $2$. Stronger tail-restricted growth conjecture. For every fixed integer $ngeq2$ 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 authors describe this as a probably stronger conjecture and report only progress toward the main conjecture; no resolution is given.