The conjecture that random 4-regular graphs are globally rigid in the real line

Let $G$ be a random $4$-regular graph on $n$ vertices. Global rigidity is considered for frameworks in $\mathbb{R}$.

Random 4-regular graph rigidity conjecture. A random $4$-regular graph with $n$ vertices is globally rigid in $\mathbb{R}$ with high probability.

The preceding result rules out the analogous statement for random $3$-regular graphs. For sufficiently large degrees, the paper's methods imply global rigidity with high probability, so the conjecture concerns whether degree four is already sufficient; the smallest suitable degree is otherwise left open.