The characterization of self-bumping representations by shuffle immersions

Let $M$ be the hyperbolic $3$-manifold under consideration, let $B$ be its relevant deformation space, and let $\mathcal C$ be a non-empty collection of curves in $M$. Set

$$\hat M=M-\mathcal C.$$

Let $s$ be a shuffle immersion with respect to $\mathcal C$, and let $\hat\rho$ be a uniformization of $\hat M$. A representation $\rho$ is a point of self-bumping for $B$ if and only if there are such $\mathcal C$, $s$, and $\hat\rho$ with

$$\rho=\hat\rho\circ s_*.$$

This conjecture seeks a complete characterization of self-bumping in terms of the shuffle immersions and uniformizations constructed in the paper; the preceding results establish that representations arising from the specified construction give points of self-bumping under additional hypotheses, while the converse and the full equivalence remain open.