Signed Talaska formula for perfectly oriented networks on surfaces

Let $N=(V,E)$ be a perfectly oriented network embedded on a closed orientable surface with boundary. Let $B(N,\{x_e\}_{e\in E})$ be its boundary measurement matrix, let $I$ and $J$ be boundary index sets with $J\subseteq K$ and $|I|=|J|$, and let $\mathcal{F}_{I,J}(N)$ and $\mathcal{C}(N)$ denote the corresponding families of flows and cycles. For a flow or cycle, write $\operatorname{wt}$ for its weight.

Signed Talaska formula. There exists a map

$$\sigma:\mathcal{F}_{I,J}(N)\cup\mathcal{C}(N)\longrightarrow\{\pm1\}$$

such that

$$\Delta_{I,J}\bigl(B(N,\{x_e\}_{e\in E})\bigr)=\frac{\sum_{\mathbf{F}\in\mathcal{F}_{I,J}(N)}\sigma(\mathbf{F})\operatorname{wt}(\mathbf{F})}{\sum_{\mathbf{C}\in\mathcal{C}(N)}\sigma(\mathbf{C})\operatorname{wt}(\mathbf{C})}.$$

This conjecture extends Talaska's subtraction-free formula from networks on a disk to perfectly oriented networks on arbitrary closed orientable surfaces with boundary, where signs are required to account for the topology.