The lattice-symmetry rigidity conjecture for proper holomorphic maps between balls

Let $f: \operatorname{\mathbb{B}}^m \rightarrow \operatorname{\mathbb{B}}^M$ be a proper holomorphic map that extends to a Hölder-continuous map $\overline{\operatorname{\mathbb{B}}^m} \rightarrow \overline{\operatorname{\mathbb{B}}^M}$. Let $\operatorname{\mathsf{G}}_f$ be the symmetry group of $f$, and consider the natural projection $\operatorname{\mathsf{G}}_f \rightarrow \operatorname{\mathsf{Aut}}(\operatorname{\mathbb{B}}^m)$. Lattice-symmetry rigidity conjecture. If the image of this projection contains a uniform lattice, then there exist $\phi_1 \in \operatorname{\mathsf{Aut}}(\operatorname{\mathbb{B}}^m)$ and $\phi_2 \in \operatorname{\mathsf{Aut}}(\operatorname{\mathbb{B}}^M)$ such that

$$\phi_2 \circ f \circ \phi_1(z) = (z,0)$$

for all $z \in \operatorname{\mathbb{B}}^m$. This is presented as an equivalent restatement of the lattice-preservation conjecture, whose special case $M \leq 2m-1$ is known; the general statement remains open.