Let \(X\) denote a maximal curve over the field \({\mathbb F}_{q^2}\), and let \(p\) denote the characteristic of this field. If there exists a rational point \(P\) on \(X\) and a function \(y\in {\mathbb F}_{q^2}(X)\) with \((y)_\infty=mP\), where \(m| (q+1)\), then \textit{R.~Fuhrmann}, \textit{F.~Torres}, and the second author [J. Number Theory 67, 29--51 (1997; Zbl 0914.11036)] showed that \(X\) must be isomorphic to the nonsingular model of \(y^q+y=x^m\). If, instead, \(m| q\) and \(x\in {\mathbb F}_{q^2}(X)\) with \((x)_\infty=mP\), then they conjectured that \(X\) is isomorphic to the curve given by \(P(z)=x^{q+1}\), where \(P(z)\) is an \({\mathbb F}_p\)-linear polynomial of degree \(m\). The authors prove a weaker result: Under the additional hypothesis that the field \( {\mathbb F}_{q^2}(X)\) is a Galois extension of \( {\mathbb F}_{q^2}(x)\), they show that in this case the curve is isomorphic to a curve given by \(P(z)=A(x)\), where \(P(z)\in {\mathbb F}_{q^2}[z]\) is an additive separable polynomial of degree \(m\) and \(A(x)\in {\mathbb F}_{q^2}[x]\) is a polynomial of degree \(q+1\). In the particular case \(m=q/p\), they show, without the Galois assumption above, that \(X\) is isomorphic to the nonsingular model of the curve given by \[ \sum_{i=1}^t z^{p^{t-i}}=c\cdot x^{q+1}, \] where \(q=p^t\) and \(c^{q-1}+1=0\). This generalizes a result in characteristic two shown by the first author and \textit{F.~Torres} [Manuscr. Math. 99, 39--53 (1999; Zbl 0931.11022)]. When \(m\) is a proper divisor of \(q\) such that \(m