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 $\|df_0\|$ over all harmonic maps $f:\mathbb D\to D$ with $f(0)=p$ whose differential $df_0$ at $0\in \mathbb D$ is conformal. If $f:\mathbb D\to D$ is a conformal diffeomorphism onto $D$ with $f(0)=p$, then $\|df_0\|\le M_D(p)$. 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 $F'(z)=\frac{c}{1+az+\lambda z^2}$, where $c>0$, $|\lambda|<1$, and $|a-\bar a\lambda|<1-|\lambda|^2$. This family contains strongly convex examples which are not round discs.