Permutation conjecture for a rational function on roots of unity

Let $k$ be an even integer and set $q=5^k$. Let

$$\mu_{q+1}=\{x\in {\mathbb F}_{q^2}:x^{q+1}=1\}$$

be the group of $(q+1)$-st roots of unity in ${\mathbb F}_{q^2}$. Permutation conjecture. The rational function

$$-x\left(\frac{x^2-2}{x^2+2}\right)^2$$

permutes $\mu_{q+1}$. If true, this would yield a class of permutation trinomials over finite fields through the relation between permutation polynomials over ${\mathbb F}_{q^2}$ and rational functions on $\mu_{q+1}$ described in the surrounding argument.