Alzer–Qiu's monotonicity and convexity conjecture for the elliptic-integral ratio

Let $\mathscr{K}(r)$ denote the complete elliptic integral of the first kind for $r\in(0,1)$, and define

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

Alzer–Qiu's conjecture. The function $G$ is strictly increasing and convex from $(0,1)$ onto $(3/4,1)$.

This conjecture refines the best possible exponents in bounds approximating the complete elliptic integral by $(\operatorname{arth} r)/r$. The source notes that the conjecture remained difficult to prove; no resolution is supplied here.