Monotonicity conjecture for the Möbius metric under midpoint rotation in ring domains

Let $R(r,1)$ be the annular ring with $0<r<1$, let $k$ satisfy $r<k<1$, and let $q$ satisfy

$$0<q<\min\{k-r,1-k\}.$$

For $\mu\in[0,\pi/2]$, consider the two points $qe^{\mu i}+k$ and $qe^{(\pi+\mu)i}+k$ in the ring, and let $S^1$ and $S^1(0,r)$ denote the boundary circles of the outer and inner disks, respectively. Midpoint-rotation monotonicity conjecture. For all $r<k<1$ and $0<q<\min\{k-r,1-k\}$, the supremum

$$\sup_{a\in S^1,\,b\in S^1(0,r)}|a,qe^{\mu i}+k,b,qe^{(\pi+\mu)i}+k|$$

is decreasing with respect to $\mu\in[0,\pi/2]$. This conjecture concerns the dependence of the Möbius metric on rotating a pair of antipodal points about their Euclidean midpoint in an annular ring; the supplied text gives no resolution, so its status remains open.