Vuorinen survey Problems 3.38(a)–(b): Newton iteration for the inverse Grötzsch modulus

Let $\mu:(0,1)\to(0,\infty)$ be the Grötzsch modulus function $\mu(r)=\frac{\pi}{2}\frac{\mathcal{K}(\sqrt{1-r^2})}{\mathcal{K}(r)}$, where $\mathcal{K}$ is the complete elliptic integral of the first kind. For $y>\frac{\pi}{2}$, set $x_0=\frac{1}{\cosh y}$ and define Newton's iteration for solving $\mu(x)=y$ by $x_{n+1}=x_n-\frac{\mu(x_n)-y}{\mu'(x_n)}$. Does $x_n$ converge to $\mu^{-1}(y)$ for every $y>\frac{\pi}{2}$? Moreover, for every $y>\pi$, is the sequence strictly increasing, i.e. does $x_n<x_{n+1}$ hold for all $n\ge 0$?