Gudmundsson's local and global harmonic morphism conjecture for irreducible symmetric spaces

Let $(M^m,g)$ be an irreducible Riemannian symmetric space of dimension $m\geq 2$. A harmonic morphism is a map that pulls back local harmonic functions to harmonic functions. For each point $p\in M$, consider an open neighbourhood $U$ of $p$.

Gudmundsson's conjecture. For each point $p\in M$, there exists a complex-valued harmonic morphism

$$\phi:U\rightarrow\mathbb{C}$$

defined on an open neighbourhood $U$ of $p$. If $(M,g)$ is of non-compact type, then $U$ can be chosen to be the whole of $M$.

This conjecture predicts local harmonic morphisms on every irreducible Riemannian symmetric space and global ones in the non-compact case. The supplied text presents it as a conjecture, but gives no resolution status.