The closed formula for the refined tree index

Let $\gamma_i$ be charge vectors, let $F_{{\rm tr},n}(\{\gamma_i\},z)$ be the partial tree index, let $\kappa$ be the two-centered index factor, and define

$$\beta_{kk}=\sum_{i=1}^{k-1}\gamma_{ik}.$$

Let ${\rm Sym}$ denote the symmetrization over the charge labels. The refined tree-index conjecture. The tree index is conjectured to satisfy

$$g_{\rm tr}(\{\gamma_i\},z,y)=(-1)^{n-1}(n-1)!\,{\rm Sym}\left\{F_{{\rm tr},n}(\{\gamma_i\},z)\prod_{k=2}^{n}\kappa(\beta_{kk})\right\}.$$

The formula is motivated by the explicit three-centered result and is proposed as a uniform representation for arbitrary $n$. The supplied text gives no proof or resolution, so its general status remains open.