Conjecture on localization of self-reinforced diffusions with trigonometric interaction

Let $(X_t)_{t\geqslant 0}$ solve the stochastic differential equation

$$dX_t=\sigma\,dW_t+\sum_{k=1}^{n}ka_k\int_0^t\sin(k(X_t-X_s))\,ds\,dt,\qquad X_0=x.$$

Self-reinforced diffusion conjecture. If $\sum_{k=1}^{n}k^2a_k<0$, then $X_t$ converges almost surely; if $\sum_{k=1}^{n}k^2a_k>0$, then

$$\limsup_{t\to\infty}X_t>\liminf_{t\to\infty}X_t.$$

The sign of the derivative of the interaction drift at zero is expected to determine whether the diffusion localizes or continues to fluctuate. The preceding analysis proves almost-sure convergence for the single-frequency model, while the corresponding assertion for the general self-reinforced diffusion is posed here as an open conjecture.