A High-Rank Gap Theorem for Type-I Bounded Symmetric Domains

Let $Ω_{r,s}$ and $Ω_{r',s'}$ be Type-I bounded symmetric domains with $s>r$ and $s'>r'$. Let $F:U\to M_{r',s'}(\mathbb C)$ be holomorphic, where $\overlineΩ_{r,s}\subset U$, and suppose that $F(S_{r,s})\subset S_{r',s'}$. If \[ k(s-r)\leq s'-r'kr, \] then, up t