Let \(\rho\) be a radial symmetric Riemannian metric defined in the annulus \(A(1, R)\). It was D. Kalaj who conjectured in [“Deformations of annuli on Riemann surfaces with smallest mean distortion”, arXiv:1005.5269]) that, if there exists a \(\rho\)-harmonic homeomorphism from the annulus \(A(1, r)\) onto \(A(1, R)\), then \(R\) satisfies the so-called \(\rho\)-Nitsche condition \[ r\leq\exp\left(\int\limits_{1}^{R}\frac{\rho(s)ds}{\sqrt{s^2\rho^2(s)-\alpha_0}}\right), \quad\alpha_{0}=\inf_{1\leq s\leq R}(\rho^{2}(s))s^{2}. \] The authors prove that if \(\rho(w) = |w|^{-2}\), then there exists a \(\rho\)-harmonic mapping between \(A(1, r)\) and \(A(1, R)\) if and only if the classical Nitsche condition \(r\leq R + \sqrt{R^2-1}\) holds. But the \(\rho\)-Nitsche condition becomes the classical Nitsche condition for the chosen \(\rho\); this yields the validity of the \(\rho\)-Nitsche condition. Hence the authors provide a positive answer to Kalaj’s conjecture for \(\rho(w) = |w|^{-2}\). Several related topics for mappings of finite distortion are also discussed.