  Assume that $f$ is a real $\rho$-harmonic function of the unit disk
$\mathbb{D}$ onto the interval $(-1,1)$, where $\rho(u,v)=R(u)$ is a metric
defined in the infinite strip $(-1,1)\times \mathbb{R}$. Then we prove that
$|\nabla f(z)|(1-|z|^2)\le \frac{4}{\pi}(1-f(z)^2)$ for all $z\in\mathbb{D}$,
provided that $\rho$ has a non-negative Gaussian curvature. This extends
several results in the field and answers to a conjecture proposed by the first
author in 2014. Such an inequality is not true for negatively curved metrics.
