The Furstenberg dimension conjecture on Riemannian surfaces

Let $E\subset\Omega$ be an $(s,t)$-Furstenberg set on a Riemannian surface: there is a family $\Gamma$ of local geodesics with $\dim_{\mathcal H}\Gamma\geq t$ such that $\dim_{\mathcal H}(E\cap\gamma)\geq s$ for every $\gamma\in\Gamma$, where $s\in(0,1]$ and $t\in(0,2]$. Furstenberg conjecture. One has

$$\dim_{\mathcal H}E\geq\min\Bigl\{s+t,\;\frac{3s+t}{2},\;s+1\Bigr\}.$$

This extends the Furstenberg problem from the Euclidean plane to Riemannian surfaces; the paper presents the assertion as a natural conjecture, and its general status is open.