Characterize maximal curves in the family

Determine whether every maximal curve in the family \(\mathcal C_{k,M}: y^k=x^M+x^{-M}\), for odd coprime integers \(k,M\ge3\), satisfies condition (A) \(k\mid q+1\) and \(M\mid q-1\), or condition (B) \(kM\mid q+1\), of Proposition 5.4.

Background

Proposition 5.4 gives two sufficient conditions under which the curves $\mathcal C_{k,M}$ are covered by the Hermitian curve and are therefore maximal over the relevant finite field. The paper explicitly notes that these conditions are independent and demonstrates that weakening either one can fail.

The unresolved issue is whether the two sufficient conditions are also exhaustive for maximal members of the family. The example $(k,M)=(3,341)$ over $\mathbb F_{1024}$ shows that merely having $k\mid q+1$ and $M\mid q^2-1$ is insufficient, but it does not settle the stated classification question.