Labourie's vector-bundle conjecture for Hitchin components

Let $S$ be a closed surface of genus $g$, let $\mathcal{M}(S)$ be its moduli space of Riemann surfaces, and let $\mathcal{T}_H(S,\operatorname{PSL}(n,\mathbf{R}))$ be the Hitchin component. For a chosen conformal structure, Hitchin's parametrization identifies it with holomorphic differentials, and the mapping class group acts on the component.

Labourie's conjecture. The quotient of $\mathcal{T}_H(S,\operatorname{PSL}(n,\mathbf{R}))$ by the mapping class group is a holomorphic vector bundle over $\mathcal{M}(S)$, with fiber equal to

$$\sum_{i=3}^{n} H^0(S,K^{i}).$$

The conjecture refines the relation between Hitchin components and moduli spaces of Riemann surfaces. The source attributes it to Labourie and does not state a resolution.