The Bochner conjecture for harmonic maps into NPC metric spaces

Let $M$ be a Riemannian manifold with non-negative Ricci curvature, and let $X$ be an NPC metric space. A harmonic map is a map $u:M\rightarrow X$ satisfying the harmonic-map condition for metric-space-valued maps. Bochner conjecture. Any harmonic map $u:M\rightarrow X$ is totally geodesic. The classical Bochner formula proves the analogous conclusion for maps into non-positively curved smooth targets, but the methods described here do not control the curvature-dependent error terms on a non-flat domain; establishing total geodesicity in the metric-space setting therefore remains open.