The regular-metric Nitsche bound for harmonic annulus maps

Let $\rho$ be a regular metric, let $A'=A(r,1)$ with $r<1$, and let $A=A(b4,c3)$. Suppose there exists a $\rho$-harmonic mapping of $A'$ onto $A$.

Nitsche bound. Then

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

This bound is attributed in the source to the cited prior work and is presented as a conjectural candidate here; the supplied text does not establish its resolution.