Li–Qu–Li–Fu's rational-function permutation conjecture on roots of unity

Let $k$ be a positive integer, set $q=3^k$, and define

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

the set of $(q+1)$-th roots of unity in ${\mathbb F}_{q^2}$. A rational function permutes $\mu_{q+1}$ if it is defined on this set and induces a bijection of it. Li–Qu–Li–Fu's rational-function conjecture. The following assertions hold:

1. If $k$ is even and

$$g(x)=\frac{-x^7+x^6+x}{x^6+x-1},$$

then $g(x)$ permutes $\mu_{q+1}$. 2. If

$$g(x)=\frac{x^6+x^4-1}{-x^7+x^3+x},$$

then $g(x)$ permutes $\mu_{q+1}$. 3. If

$$g(x)=\frac{-x^5+x^3+x}{x^4+x^2-1},$$

then $g(x)$ permutes $\mu_{q+1}$ whenever $k\not\equiv2\pmod{4}$.

These assertions are presented as an equivalent reformulation of the preceding permutation-trinomial conjecture via the stated criterion for permutation polynomials. The source context gives no resolution of these assertions.