An upper bound for an exceptional automorphism group

Let $q=p^h>7$ be odd, put $m=(q+1)\mathrm{/}2$, and suppose that $i=(m-2)\mathrm{/}2$ satisfies $\gcd (i,m)=\gcd (i+2,m)=1$. For the ${\mathbf{F}}_{q^2}$-maximal function field ${\mathcal{F}}_i={\mathbf{F}}_{q^2}(x,y)$ defined by $y^m=x^i(x^2+1)$, Peter Beelen, Maria Montanucci, Jonathan Niemann, and Luciane Quoos showed that the geometric automorphism group contains a subgroup of order $4(q+1)$ and conjectured that its order is exactly $4(q+1)$. We prove this equality by establishing the reverse inequality.