The generic spectrum conjecture for SL(2,R)-representations of closed surfaces

Let $\Sigma_g$ be a closed surface of genus $g \geq 2$, and let $\rho: \Sigma_g \longrightarrow \mathrm{SL}(2,\mathbb{R})$ be a representation. The spectrum of $\rho$ is the subset of measured laminations defined by limits of simple closed curves along which the spectral radius of $\rho$ grows subexponentially. A representation is simple Fuchsian when it has that property in the sense of the paper.

Generic spectrum conjecture. For a generic $\rho: \Sigma_g \longrightarrow \mathrm{SL}(2,\mathbb{R})$ which is not simple Fuchsian, the spectrum of $\rho$ is closed and has empty interior, and locally has the structure of the product of a Cantor set by a codimension $1$ manifold.

The conjecture extends the paper's genus-one results to closed surfaces of genus at least two and predicts a detailed local structure for the spectrum. Its status is open; the source describes this as a general conjecture suggested by the results.