Anderson–Vamanamurthy–Vuorinen elliptic-integral monotonicity conjecture

Let $\mathcal{K}(r)=(\pi/2)F(1/2,1/2;1;r^2)$ be the complete elliptic integral of the first kind. For $r\in(0,1)$, set $r'=\sqrt{1-r^2}$. Anderson–Vamanamurthy–Vuorinen's conjecture. The function

$$r\mapsto\frac{\mathcal{K}(r)/\ln(4/r')-1}{(r')^2}$$

is increasing from $(0,1)$ onto $(\pi/\ln 16-1,1/4)$. This is a special elliptic-integral instance of the broader open-problem context surrounding Ramanujan's asymptotic formula for the zero-balanced hypergeometric function; the supplied text does not establish whether this particular conjecture has been resolved.