Zariski-density conjecture for intermediate components of SO(n,n+1)-character varieties

Let $X$ be a closed Riemann surface of genus $g$, let $n\geq 3$, and let ${\mathcal X}_d({\mathsf {SO}}(n,n+1))$ be the component of the ${\mathsf {SO}}(n,n+1)$-character variety described in Theorem 1, with $0<d<n(2g-2)$. Zariski-density conjecture. Every representation in

$${\mathcal X}_d({\mathsf {SO}}(n,n+1))$$

is Zariski dense. The components in question are smooth and hence consist of irreducible representations; the conjecture asserts that, apart from the boundary cases, none has a proper algebraic Zariski closure. The source gives no resolution of this conjecture.