Neumann's surjectivity conjecture for the Euler-class parametrization

Let $\Sigma$ be a closed surface, let $G=\mathsf{PSL}(2,\mathbb{R})$, and let

$$\mathbb{U}:\mathfrak{S}^k(\Sigma)\longrightarrow e^{-1}(\chi(\Sigma)+k)$$

be the natural map from the holomorphic symmetric-power bundle to the corresponding Euler-class component. A $\mathsf{PSL}(2,\mathbb{R})$-representation has dense image if its image in $\mathsf{PSL}(2,\mathbb{R})$ is dense.

Neumann's surjectivity conjecture. If $k=1$, then $\mathbb{U}$ is onto. In general, every $\mathsf{PSL}(2,\mathbb{R})$-representation with dense image lies in $\operatorname{Image}(\mathbb{U})$.

The conjecture addresses the failure of the parametrization map to be onto in general, while asserting surjectivity for $k=1$ and inclusion of all dense-image representations in its image. The source attributes the conjecture to discussions with Neumann and gives no resolution.