The Bernardi-process conjecture for determining ribbon-graph genus

Let $(G,\rho)$ be a ribbon graph. Write $V(G)$ for its vertices, $E(G)$ for its edges, $\operatorname{Pic}^0(G)$ for its degree-zero Picard group, and $\mathcal{T}(G)\subset E(G)$ for its spanning trees. For every $v\in V(G)$, suppose we are given the map

$$\operatorname{Pic}^0(G)\times\mathcal{T}(G)\xrightarrow{\beta_v(\operatorname{Pic}^0(G))}\mathcal{T}(G),$$

where $\beta_v$ is the Bernardi process with basepoint $v$. Bernardi-process conjecture. It is possible to determine the genus of $(G,\rho)$ from this data. The challenge is that, even for a cut-free graph, the Bernardi process does not readily detect the cyclic order around a fixed vertex without information about the cyclic orders around other vertices.