Wu–Li's permutation conjecture on $5$-power roots of unity

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

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

Define $g(x)=-x\big(\frac{x^2-2}{x^2+2}\big)^2$. Wu–Li's conjecture. The map $g$ permutates $\mu_{q+1}$. This is the second of two conjectures proposed by Wu and Li to obtain new permutation trinomials; the paper's abstract states that both conjectures are settled.