Existence conjecture for harmonic morphisms from irreducible Riemannian symmetric spaces

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

Existence conjecture. There exists a complex-valued harmonic morphism

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

for each point $p\in M$. If $(M,g)$ is of non-compact type, 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 paper gives positive answers for several specified symmetric spaces and proves global existence for spaces of type IV, but the full assertion is not established in the supplied text.