Gudmundsson–Svensson conjecture on local and global harmonic morphisms from irreducible symmetric spaces

Let $(M^m,g)$ be an irreducible Riemannian symmetric space of dimension $m\ge 2$. For each point $p\in M$, a complex valued harmonic morphism is a harmonic morphism

$$\phi:U\to\mathbb C$$

defined on an open neighbourhood $U$ of $p$. Gudmundsson–Svensson conjecture. For each point $p\in M$, there exists a complex valued harmonic morphism $\phi:U\to\mathbb C$ defined on an open neighbourhood $U$ of $p$. If the space $(M,g)$ is of non-compact type, then the domain $U$ can be chosen to be the whole of $M$. The conjecture proposes a general local existence result for complex valued harmonic morphisms on irreducible Riemannian symmetric spaces, together with global existence in the non-compact case. The abstract states that the corresponding existence problem has a positive answer in the real and quaternionic Grassmannian cases, while the conjecture is presented as the broader motivating claim.