Rational-point formulation of Helleseth's conjecture for the curves $D_k$

Let

$$D_k: y^2+y=x^{2^k+1}+x^{-1}$$

be the curve over ${\mathbb F}_2$, and let $k,m$ be positive integers.

Rational-point formulation. When $\gcd(k,m)=1$, the curves $D_1$ and $D_k$ have the same number of rational points over ${\mathbb F}_{2^m}$:

$$\#D_1({\mathbb F}_{2^m})=\#D_k({\mathbb F}_{2^m}).$$

The paper derives this as an equivalent reformulation of Helleseth's exponential-sum conjecture using the identity $\#D_k({\mathbb F}_{2^m})=2^m+1-G_m^{(k)}$. The supplied text gives no resolution status.