The gcd-dependence conjecture for Gold exponent exponential sums

Let $m$ be a positive integer and, for a positive integer $k$, define the exponential sum

$$G_m^{(k)}=\sum\limits_{x\in \operatorname{GF}(2^m)^*}(-1)^{\operatorname{Tr}_m(x^{2^k+1}+x^{-1})}.$$

Gcd-dependence conjecture. For any positive integer $k$,

$$G_m^{(k)}=G_m^{(\operatorname{gcd}(k,m))}.$$

Equivalently, $G_m^{(k)}$ depends only on $\operatorname{gcd}(k,m)$; in particular, if $\operatorname{gcd}(k,m)=1$, then

$$G_m^{(k)}=\sum\limits_{x\in \operatorname{GF}(2^m)^*}(-1)^{\operatorname{Tr}_m(x^3+x^{-1})}.$$

This conjecture is one of two claims on exponential sums used in the proposed determination of the five-valued cross-correlation distribution for the case $d=(2^{2k}+1)/(2^k+1)$. The source provides no resolution, and the conjecture remains open here.