Permutation criterion for two families of pentanomials over finite fields

Let $m$ and $k$ be positive integers and set $q=2^m$. Define

$$f_1(x)=x^{2^k+1}+x^{2^kq^2+1}+x^{q+2^k}+x^{q^2+2^k}+x^{2^kq+1}$$

and

$$f_2(x)=x^{2^k+1}+x^{(2^k+1)q}+x^{q+2^k}+x^{q^2+2^k}+x^{2^kq^2+q}.$$

Permutation criterion. The polynomials $f_1$ and $f_2$ are permutation polynomials over $\mathbb{F}_{q^3}$ if and only if

$$\operatorname{gcd}(2^k+1,q-1)=1.$$

This conjecture is based on numerical results concerning permutation pentanomials over finite fields of characteristic two; the supplied text does not establish whether the criterion has been proved or disproved.