Universal building conjecture for spectral covers

Let $X$ be a Riemann surface and let $\phi$ be a smooth spectral cover of $X$. A universal harmonic $\phi$-map is a harmonic $\phi$-map $h^{\phi}:\widetilde X\to\mathcal B^{\phi}$ such that every harmonic $\phi$-map from $\widetilde X$ to a building with a complete system of apartments and the same vectorial Weyl group factors through a folding map from $\mathcal B^{\phi}$, uniquely on the image. Universal building conjecture. There exists a universal $\phi$-map

$$h^{\phi}:\widetilde X\longrightarrow\mathcal B^{\phi}.$$

For rank two the universal object is the leaf-space $\mathbb R$-tree of the induced foliation, while in totally decomposed and simple-branch cases the paper describes explicit simpler models. Existence in general is presented as one of the paper's main conjectures and remains open.