The thin elements conjecture for the branching semi-globular nerve

Let $\mathcal{C}$ be a non-contracting $\omega$-category freely generated by a precubical set. Let $M_n^{gl^-}(\mathcal{C})$ denote the degree-$n$ elements of the branching semi-globular complex, and let $\sum_{i\in I}\lambda_i x_i$ be an integral linear combination with $\lambda_i\in\mathbb{Z}$. The branching semi-globular thin elements conjecture. The linear combination $\sum_{i\in I}\lambda_i x_i$ is a cycle if and only if it is a boundary for the simplicial differential map. This is the corresponding thin-elements assertion for the branching semi-globular nerve; the source gives no evidence of resolution.