Durrett–Rogers law of large numbers conjecture for Brownian polymers

Let $(X(t))_{t\ge0}$ be the process defined by

$$X(t)=B(t)+\int_0^t\int_0^s\left(\xi(X(s))+f(X(s)-X(u))\,du\right)ds,$$

where $B(t)$ is standard one-dimensional Brownian motion, $f:\mathbb R\to\mathbb R$ has sufficient regularity and sufficiently fast decay at infinity, and $\xi$ is an initial drift profile. Assume

$$f(-x)=-f(x),\qquad \operatorname{sgn}(f(x))=\operatorname{sgn}(x).$$

Durrett–Rogers law of large numbers conjecture. Under these assumptions, $X(t)/t\to0$ almost surely. This conjecture concerns the long-time displacement of a one-dimensional self-interacting Brownian polymer; its status is not resolved in the supplied source.