Nonzero limiting endpoint distance for hyperbolic self-avoiding walks at a power-law step size

Consider continuous $n$-step self-avoiding walks on the hyperbolic space $\mathbb{H}^d$, with steps of length $\varepsilon$ and with the distance between every pair of vertices conditioned to exceed $c\varepsilon$, where $0<c<1$. Write $\mathbb{E}_{n,\varepsilon}$ for expectation under this measure. Power-law scaling conjecture. There exists a constant $\beta=\beta(d,c)$ with $0<\beta<1$ such that

$$\lim_{n\to\infty}\mathbb{E}_{n,n^{-\beta}}\bigl(d(x_0,x_n)\bigr)$$

exists and is not zero, possibly up to logarithmic factors. This conjecture proposes a nonzero limiting endpoint distance in an intermediate power-law scaling regime, complementing the inverse-linear step-size problem; its resolution is left open in the source.