Sublinear local time under the recurrent limiting measure

Let $f$ satisfy

$$\int_1^\infty \frac{f(t)}{t^{3/2}}\,dt=\infty,$$

and let $\mathbb{P}$ be any weak limit of the conditioned Wiener measures $\mathbb{W}_t$. Let $L_t$ denote the local time at the origin at time $t$. Local-time separation conjecture. There exists a nonnegative deterministic function $\omega(t)$ with $\omega(t)\to\infty$ as $t\to\infty$ such that

$$\mathbb{P}\left(L_t\le\frac{f(t)}{\omega(t)}\right)\to1$$

as $t\to\infty$. This is proposed in the recurrent regime as a further description of the limiting process and is presented as an open problem.