Wu–Li's permutation-polynomial conjecture for odd extensions of $5$

Let $k$ be a positive odd integer and let $f(x)=x\big(\frac{x^2-x+2}{x^2+x+2}\big)^2$ over the finite field $\mathbb{F}_{5^k}$. Wu–Li's conjecture. The polynomial $f$ is a permutation polynomial of $\mathbb{F}_{5^k}$. This is one of two conjectures proposed by Wu and Li concerning permutation trinomials over finite fields; the paper states that both conjectures are settled.