Tree-symmetrization identity for the Joyce potential

Let $\mathcal{T}$ be an unrooted, unlabeled tree with $n$ vertices, and let $K_\mathcal{T}(\{\gamma_i,t_i\})$ be the tree kernel defined by the product over its oriented edges. Write ${\rm Sym}$ for symmetrization over the labels. For each labeling $\ell:v\mapsto i\in[1,n]$ of the vertices, orient every edge from $v$ to $v'$ whenever $\ell(v)<\ell(v')$. Tree-symmetrization conjecture. For each unrooted, unlabeled tree $\mathcal{T}$ one has

$${\rm Sym}\,K_\mathcal{T}(\{\gamma_i,t_i\})=\frac{1}{(4\pi\mathrm{i})^{n-1}}\,{\rm Sym}\,\Biggl[\left(\sum_{\rm labels}\prod_{e:i\to j}\langle\gamma_i,\gamma_j\rangle\right)\prod_{i=1}^{n-1}\frac{t_i t_{i+1}}{t_{i+1}-t_i}\Biggr].$$

The identity is proposed to establish that the tree expansion of the Fourier modes of the Joyce potential agrees with the expression obtained from the Joyce construction, thereby relating the factorized and tree-based descriptions of the heavenly metric.