Necessary lower bound for radial harmonic mappings of annuli with regular metrics

Let $\rho$ be a regular metric, let $A'=A(r,1)$ with $r<1$, and let $A=A(\tau,\sigma)$ with $\tau\leq\sigma\leq 1$. A $\rho$-harmonic mapping from $A'$ onto $A$ is assumed to exist.

The regular-metric annulus conjecture. Under these hypotheses,

$$r\geq \exp\left(\int_{\sigma}^{\tau} \frac{\rho(y)\,dy}{\sqrt{y^2\rho^2(y)-\tau^2\rho^2(\tau)}}\right).$$

This gives a necessary lower bound for the inner radius of the source annulus in terms of the regular metric and the target annulus. The supplied text presents it in a conjecture environment, and no resolution is given in the source span.