Rigidity conjecture for proper maps between type-I bounded symmetric domains

Let $D^{\mathrm{I}}_{p,q}$ denote the irreducible type-I bounded symmetric domain, and let $F:D^{\mathrm{I}}_{p,q}\to D^{\mathrm{I}}_{p',q'}$ be a proper holomorphic map, where $p\ge q\ge 2$ and $q'<p$. A map is of diagonal type if it is equivalent to a map $G_h$ of the form

$$G_h(Z)=\begin{bmatrix} Z & {\bf 0}\\ {\bf 0} & h(Z)\end{bmatrix},$$

for some holomorphic map $h:D^{\mathrm{I}}_{p,q}\to D^{\mathrm{I}}_{p'-p,q'-q}$. Rigidity conjecture. If $p'<2p-1$ or $q'<2q-1$, then $p\le p'$, $q\le q'$, and $F$ is of diagonal type. Motivated by rigidity results of Kim–Zaitsev and Kim, this predicts that proper holomorphic maps in the indicated rank-at-least-two range have no additional forms under either numerical bound; the status is not established in the supplied source.