Walk dimension conjecture for fractional Brownian motion graphs

Let $B^H$ be a fractional Brownian motion with Hurst index $H\in(0,1)$, and let $\varphi(W)$ be the Hunt process on $\operatorname{gr}(B^H)$ induced by a Wiener process through the graph map $\varphi$. Write $\dim_{\mathcal{W}}$ for the walk dimension of this process. Walk dimension conjecture. With probability one,

$$\dim_{\mathcal{W}}(\operatorname{gr}(B^H),\varphi(W))=\frac{2}{H}.$$

The graph of fractional Brownian motion has Hausdorff dimension $2-H$, while the corresponding spectral dimension is $1/2$; this conjecture predicts the remaining walk dimension for the induced process.