The sharp quotient bounds conjecture for the Möbius metric in sectors

Let $Q(k,\theta)$ denote the quotient studied in the paper, with $0<k<1$ and $0<\theta<2\pi$. The sharp quotient bounds conjecture. For all such $k$ and $\theta$,

$$1=\lim_{k\to1^-}Q(k,\theta)\leq Q(k,\theta)\leq\lim_{k\to0^+}Q(k,\theta)=\frac{\theta\sin(\theta/2)}{2\pi\sin^2(\theta/4)},\qquad \theta<\pi,$$

and

$$\frac{\theta\sin(\theta/2)}{2\pi\sin^2(\theta/4)}=\lim_{k\to0^+}Q(k,\theta)\leq Q(k,\theta)\leq\lim_{k\to1^-}Q(k,\theta)=1,\qquad \theta>\pi.$$

The conjecture is motivated by numerical evidence that $Q(k,\theta)$ is monotone in $k$, decreasing for $\theta\leq\pi$ and increasing for $\theta\geq\pi$; its resolution is not stated.