Rosen's fourth-power central limit conjecture for Brownian local time increments

Let $B=(B_t)_{t\geq 0}$ be Brownian motion, let $L_t^x$ denote its local time at $x\in\mathbb{R}$, and define the increment

$$\Delta_x^hL_t^x=L_t^{x+h}-L_t^x.$$

Let $Z\sim\mathcal{N}(0,1)$ be independent of $(L_t^x)_{x\in\mathbb{R}}$. Rosen's conjecture. It holds that

$$\frac{1}{h^{5/2}}\left(\int_{-\infty}^{\infty}(\Delta_x^hL_t^x)^4\,\operatorname{d}x-24h\int_{-\infty}^{\infty}(\Delta_x^hL_t^x)^2L_t^x\,\operatorname{d}x+48h^2\int_{-\infty}^{\infty}(L_t^x)^2\,\operatorname{d}x-\int_{-\infty}^{\infty}(\Delta_x^hL_t^x)L_t^x\,\operatorname{d}x\right) \xrightarrow{d} \sqrt{\frac{2^{9}4!}{5}\int_{-\infty}^{\infty}(L_t^x)^4\,\operatorname{d}x}\,Z.$$

This extends the established central limit theorems for lower powers of Brownian local-time increments; the source explains that Rosen's method-of-moments proof for the third power does not extend to powers higher than three, so the displayed fourth-power limit remains conjectural.