Sharpness of the conformal-radius and hyperbolic-distance bounds in rectangles

Let $R=[-k,k]\times[-1,1]$ be a rectangle with $k\geq 1$, and let $s_R(u,v)$ and $\rho_R(u,v)$ denote the quantities defined in the source. Let $C(\lambda)$ be defined by

C(\lambda)=\mathchoice % %%%displaystyle %{\text{\,\fFt K}} %%%%%textstyle %{\text{\,\fFt K}}Remark %%%%scriptstyle %{\text{\,\fFa K}} %%%%%scriptscriptstyle %{\text{\,\fFp K}}(\lambda)\frac{\sqrt{(1+a^2(\lambda))(1+\lambda^2a^2(\lambda))}}{a(\lambda)},\quad \text{where}\quad a(\lambda):=|\mathrm{sn}\,(i\mathchoice % %%%displaystyle %{\text{\,\fFt K}} %%%%%textstyle %{\text{\,\fFt K}}Remark %%%%scriptstyle %{\text{\,\fFa K}} %%%%%scriptscriptstyle %{\text{\,\fFp K}}(\lambda),\lambda)|.

Rectangle inequality. For all points $u,v\in R$,

$$s_R(u,v)\leq \mathrm{th}\,\frac{\rho_R(u,v)}{2}\leq C(\lambda)s_R(u,v).$$

These inequalities are sharp, giving optimal two-sided comparison constants for the relevant metrics in rectangles.