The explicit contraction factor conjecture for Chebyshev subdivision

Let $0<\alpha<1$ and let $0\leq\beta\leq 1-\alpha$. Write $\tau_{\alpha,\beta}$ for the asymptotic degree ratio associated with approximating a Chebyshev polynomial after the affine map $x\mapsto\alpha x+\beta$. Explicit contraction factor conjecture.

$$\tau_{\alpha,\beta}=\frac{1}{(\frac{1}{\alpha}-\frac{1}{\sqrt{\alpha}})\sqrt{1-(\frac{\beta}{1-\alpha})^2}+\frac{1}{\sqrt{\alpha}}}.$$

The formula would provide an explicit value for the contraction factor whose general value is otherwise unknown; the stated bounds $\tau_{\alpha,0}\leq\alpha$ and $\tau_{\alpha,1-\alpha}\leq\sqrt{\alpha}$ are consistent with the endpoint cases, while the conjecture is supported by numerical testing.