The Nitsche inequality conjecture for $(\rho,n)$-harmonic mappings between annuli

Let $A(1,R)$ and $A(1,R_\ast)$ be annuli, and let $\rho$ be the weight defining $(\rho,n)$-harmonicity. For an increasing radial diffeomorphism between them, write

$$\mathcal{L}[H]=\rho[H(t)]\left(H(t)^2-t^2\dot H^2\right)\left(H(t)^2+\frac{t^2\dot H^2}{n-1}\right)^{\frac{1}{2}(n-2)}\equiv c,$$

and set $c_\diamond=\rho(1)$. Nitsche inequality conjecture. There is a $(\rho,n)$-harmonic mapping between annuli $A(1,R)$ and $A(1,R_\ast)$ if and only if

$$c\leq c_\diamond=\rho(1).$$

In the special case $\rho\equiv 1$ and $n=2$, this becomes the classical Nitsche inequality $R_\ast\geq (1+R^2)/(2R)$. The claim proposes that the necessary inequality derived for radial solutions is also sufficient, extending the planar harmonic annulus criterion to $(\rho,n)$-harmonic mappings.