The continuous-circle logarithmic greedy-routing conjecture

Let $n$ points be drawn independently and uniformly from the unit circle $\mathbb{S}^1$, or from the interval $[0,1]$ with periodic boundary conditions. Insert the points sequentially, connecting each new point to its $K$ nearest already-inserted neighbors. Choose $s$ and $t$ uniformly at random from the $n$ points.

Continuous-circle greedy-routing conjecture. For every $\varepsilon>0$, there exists a constant $K_0=K_0(\varepsilon)$ such that, for all $K\geq K_0$ and sufficiently large $n$, with probability at least $1-\varepsilon$, the greedy walk from $s$ reaches $t$ and completes in at most

$$C(\varepsilon)\log n$$

steps, where $C(\varepsilon)$ depends only on $\varepsilon$.

This conjecture extends the proved one-dimensional lattice result to a directionless continuous model. It is motivated by empirical evidence from growth-based approximate-nearest-neighbor structures and remains open in the supplied paper.