The period-function monotonicity and endpoint-limit conjecture for $G_\alpha$

Let $P(\beta)$ be the period function in $G_\alpha$, with $\beta$ the parameter used to specify the initial tangent vector. The period-function conjecture.

$$\frac{d}{d\beta}P(\beta)<0$$

and

$$\lim_{\beta\to1}P(\beta)=\frac{\pi\sqrt{2}}{\sqrt{\alpha}}.$$

The source establishes the corresponding monotonicity and endpoint limits for $\alpha=1$ and $\alpha=1/2$, while numerical computations support the proposed formula more generally. If true, the conjecture would give quantitative information about how the cut-locus behavior changes across the family.