Monotonicity conjecture for the normalized elliptic-integral approximation error

Let $\mathscr{K}(r)$ be the complete elliptic integral of the first kind, let

$$G(r)=\frac{\log(2\mathscr{K}(r)/\pi)}{\log((\operatorname{arth} r)/r)},$$

and define

$$f(r)=\frac{1}{r^2}\left[G(r)-\frac34\right],\qquad r\in(0,1).$$

Monotonicity conjecture for $f$. The function $f$ is strictly increasing from $(0,1)$ onto $(1/320,1/4)$.

If true, this would identify the sharp constants in the refined two-sided approximation with exponents $3/4+\alpha r^2$ and $3/4+\beta r^2$. The statement is presented as a conjecture based on earlier bounds and computation; its resolution is not given in the supplied text.