On an extremal problem for harmonic maps conformal at a point

Let $\mathbb{D}$ denote the unit disc in $\mathbb{C}$. For a domain $D\subset \mathbb{C}$ and a point $p\in D$, let $M_D(p)$ denote the supremum of $\mathrm{\parallel }d{f}_0\mathrm{\parallel }$ over all harmonic maps $f:\mathbb{D}\to D$ with $f(0)=p$ whose differential $d{f}_0$ at $0\in \mathbb{D}$ is conformal. If $f:\mathbb{D}\to D$ is a conformal diffeomorphism onto $D$ with $f(0)=p$, then $\mathrm{\parallel }d{f}_0\mathrm{\parallel }\le M_D(p)$. In a recent paper, the authors proved that equality holds when $D=\mathbb{D}$, and they asked whether equality can hold only when $D$ is a round disc. We give a negative answer by proving that, among bounded convex pointed domains $p\in D\subset \mathbb{C}$ and up to translations, rotations, and reflections, equality holds if and only if, after moving $p$ to the origin, $D=F(\mathbb{D})$ where $F:\mathbb{D}\to \mathbb{C}$ is a holomorphic map with $F(0)=0$ and $F^{\mathrm{\prime }}(z)=\frac{c}{1+az+\lambda z^2}$, where $c>0$, $\mathrm{\mid }\lambda \mathrm{\mid }<1$, and $\mathrm{\mid }a-\bar{a}\lambda \mathrm{\mid }<1-\mathrm{\mid }\lambda {\mathrm{\mid }}^2$. This family contains strongly convex examples which are not round discs.