Conjecture on dyadic local time for fractional Brownian motion

Let $B$ be fractional Brownian motion with Hurst parameter $H\in(0,1)$, and let $(\pi_n)$ be the dyadic Lebesgue partition generated by $B$. Let $I^n_k$ and $U_t$ denote the dyadic intervals and upcrossing counts from the preceding lemma, with upcrossings of $B$. Define

$$\widetilde{L}^{\pi_n}_t(x):=\sum_{k\in\mathbb{Z}}2^{-n(1/H-1)}U_t(I^n_k)\mathbf{1}_{I^n_k}(x).$$

The dyadic local-time conjecture. Almost surely, $\widetilde{L}^{\pi_n}_t(x)$ converges uniformly for $(t,x)\in[0,T]\times\mathbb{R}$ to

$$\ell_t(x)\frac{\mathbb{E}[|B_1|^{1/H}]}{2},$$

where $\ell$ is the local time of $B$, namely the Radon–Nikodym derivative of the occupation measure $A\mapsto\int_0^t\mathbf{1}_A(B(s))\,\mathrm{d}s$ with respect to Lebesgue measure. In particular, for every even integer $p\in2\mathbb{N}$, $B\in\mathcal{L}^{p-1}_q(\pi_n)$ for every $q\in(1,\infty)$. This would identify the higher-order local-time approximation from dyadic upcrossings with the usual occupation local time and imply the stated variation property for fractional Brownian motion; the supplied text gives no resolution status.