The rooted-tree and oriented-tree correspondence in Joyce–Song wall-crossing

Let $\mathcal{T}'$ be a $\Gamma$-labelled tree. In the Gaiotto–Moore–Neitzke asymptotic expansion, consider the total contribution to wall crossing obtained by choosing every possible root of $\mathcal{T}'$, and in the Joyce–Song wall-crossing formula, consider the total contribution obtained from every possible orientation of $\mathcal{T}'$. Root–orientation correspondence conjecture. These two total contributions match. The statement concerns the conjectural identification of the asymptotic expansion with the Joyce–Song wall-crossing formula. In general, the Gaiotto–Moore–Neitzke contribution is only well defined up to certain singular integrals, so the conjecture also requires a corresponding assertion about the behaviour of those integrals; the paper proves the procedure in a number of examples.