The non-deterministic VG algorithm for reconstructing oriented matroids

Let $\mathcal{A}=\{H_1,\dots,H_n\}$ be a real arrangement in $\mathbb{R}^\ell$. A generalized tope graph $\operatorname{\widetilde{\mathcal{T}}}(y_1,\dots,y_n)$ is the graph constructed from generalized Heaviside elements $y_i\in\widetilde{\mathcal{H}}(\mathcal{A})$ as in the paper. Non-deterministic reconstruction. Choose $y_1,\dots,y_n$ so that $1,y_1,\dots,y_n$ form a basis of $\operatorname{\mathsf{F}}^1\operatorname{\mathcal{VG}}(\mathcal{A})$; if the resulting generalized tope graph is the tope graph of some oriented matroid, stop, and otherwise repeat with a different choice. Then the resulting graph $\operatorname{\widetilde{\mathcal{T}}}(y_1,\dots,y_n)$ is isomorphic to the tope graph $\operatorname{\mathcal{T}}(\mathcal{A})$. This is a proposed reconstruction procedure for arbitrary real arrangements, but the supplied text gives no resolution of its conjectural guarantee.