The midpoint-coordinate limit conjecture for $G_\alpha$

For $\alpha\in(0,1]$, define

$$L(\alpha):=\lim_{x_0\to1}b_{x_0,\alpha}\bigl(P(x_0,\alpha)/2\bigr).$$

Here $b_{x_0,\alpha}$ is the coordinate function and $P(x_0,\alpha)$ is the associated period. The midpoint-coordinate limit conjecture. $L(\alpha)$ is monotonically decreasing from $\alpha=0$ to $\alpha=1$, $\lim_{\alpha\to0}L(\alpha)=\infty$, and in fact

$$L(\alpha)=\frac{2}{\alpha}.$$

The value $L(1/2)=4$ is proved in the source, while the broader monotonicity, divergence, and explicit formula are conjectural.