The conjecture on nontrivial knots and parabolic representations

Let $K$ be a knot, and let $8par(K)$ denote the space of parabolic representations associated with $K$.

Parabolic-representation existence conjecture. A knot $K$ is nontrivial if and only if $8par(K)$ is nonempty. Moreover, $8par(K)$ has a zero-dimensional component.

The conjecture addresses the existence of non-abelian parabolic representations of knot groups. Existence is known for irreducible $SU(2)$-representations of every nontrivial knot, but the corresponding general parabolic-representation question is presented as unresolved.