Huang–Ji–Yin gap-rigidity conjecture for proper rational maps between balls

Let $n\ge 3$ be a positive integer. Write $K(n)$ for the largest positive integer $m$ such that

$$n>\frac{m(m+1)}{2}.$$

For each $k$ with $1\le k\le K(n)$, let $\mathcal{I}_k$ be the collection of integers $m'$ satisfying

$$kn<m'<(k+1)n-\frac{k(k+1)}{2}.$$

Write $\operatorname{Rat}(\mathbb{B}^n,\mathbb{B}^N)$ for the proper holomorphic rational maps from $\mathbb{B}^n$ to $\mathbb{B}^N$, and call two such maps equivalent if they differ by automorphisms of the source and target balls. Huang–Ji–Yin gap-rigidity conjecture. If $N\in\mathcal{I}_k$ for some $1\le k\le K(n)$, then every proper holomorphic rational map $F\in\operatorname{Rat}(\mathbb{B}^n,\mathbb{B}^N)$ is equivalent to a map $(G,0)$, where $G$ is a proper holomorphic rational map from $\mathbb{B}^n$ into $\mathbb{B}^{N'}$ with $N'=kn<N$. This conjecture predicts a gap in the target dimensions of proper rational maps between complex balls, identifying maps in the specified intervals with lower-dimensional maps after adding zero components; its resolution concerns the rigidity and classification of proper holomorphic maps between balls.