Uniqueness conjecture for global generalized Mumford–Shah minimizers with one unbounded component

Let $(K,u)$ be a global generalized minimizer of $E_\lambda$ such that $K$ has one unbounded connected component. An elementary global minimizer and a cracktip are the two classes of global generalized minimizers described in the paper.

Uniqueness conjecture. The pair $(K,u)$ is either an elementary global minimizer or a cracktip.

This conjecture would strengthen the known structural results and would help exclude accumulation of infinitely many connected components at terminal points of connected components of positive length. Its status is not resolved in the supplied material.