Li–Qu–Li–Fu's permutation trinomial conjecture over finite fields

Let $k$ be a positive integer, set $q=3^k$, and let ${\mathbb F}_{q^2}$ be the finite field with $q^2$ elements. A polynomial is a permutation trinomial over ${\mathbb F}_{q^2}$ if it has three terms and induces a permutation of ${\mathbb F}_{q^2}$. Li–Qu–Li–Fu's conjecture. The following assertions hold:

1. If $k$ is even and

$$f(x)=x^{lq+l+5}+x^{(l+5)q+l}-x^{(l-1)q+l+6},$$

with $\gcd(5+2l,q-1)=1$, then $f(x)$ is a permutation trinomial over ${\mathbb F}_{q^2}$. 2. If

$$f(x)=x^{lq+l+1}-x^{(l+4)q+l-3}+x^{(l-2)q+l+3}$$

and $\gcd(1+2l,q-1)=1$, then $f(x)$ is a permutation trinomial over ${\mathbb F}_{q^2}$. 3. If

$$f(x)=x^{lq+l+1}+x^{(l+2)q+l-1}-x^{(l-2)q+l+3}$$

and $\gcd(1+2l,q-1)=1$, then $f(x)$ is a permutation trinomial over ${\mathbb F}_{q^2}$ whenever $k\not\equiv 2\pmod{4}$.

These conjectures concern explicit permutation trinomials over quadratic extensions of fields of characteristic three. The paper proposes them as finite-field permutation problems; no resolution is supplied in the given source context.