Conjectured permutation trinomials over finite fields of characteristic three

Let $k$ be a positive integer, let $q=3^k$, and let $l$ be an integer. For the first and third families, assume the stated conditions on $k$; in every family, assume the stated coprimality condition on $l$ and $q-1$. Define the following polynomials over $\mathbb{F}_{q^2}$:

$$f_1(x)=x^{lq+l+5}+x^{(l+5)q+l}-x^{(l-1)q+l+6},$$ $$f_2(x)=x^{lq+l+1}-x^{(l+4)q+l-3}+x^{(l-2)q+l+3},$$ $$f_3(x)=x^{lq+l+1}+x^{(l+2)q+l-1}-x^{(l-2)q+l+3}.$$

Permutation trinomial conjectures. (1) If $k$ is even and $\mathrm{gcd}(5+2l,q-1)=1$, then $f_1(x)$ is a permutation trinomial over $\mathbb{F}_{q^2}$. (2) If $\mathrm{gcd}(1+2l,q-1)=1$, then $f_2(x)$ is a permutation trinomial over $\mathbb{F}_{q^2}$. (3) If $\mathrm{gcd}(1+2l,q-1)=1$ and $k\not\equiv2\pmod4$, then $f_3(x)$ is a permutation trinomial over $\mathbb{F}_{q^2}$.

These are computer-supported conjectures proposing three parameterized families of permutation trinomials over finite fields of characteristic three. The source gives no proof or resolution, so their status remains open.