Durrett–Rogers conjecture on sublinear growth of repulsive self-interacting motion

Suppose that $f:\mathbb{R}\rightarrow\mathbb{R}$ is an odd function of compact support such that $xf(x)\ge 0$. Let $X$ be the process defined by the Durrett–Rogers equation. Durrett–Rogers conjecture. The quotient

$$\frac{X_t}{t}$$

converges almost surely to $0$. This conjecture concerns the long-time behaviour of a self-interacting diffusion with repulsive compactly supported interaction; it was partially solved by Tarrès, Tóth and Valkó, but the supplied source does not establish a complete resolution.