One-modulus formula for refined tree coefficients

Let $\boldsymbol\gamma=(\gamma_1,\ldots,\gamma_n)$ be an ordered collection of charges, let $\hat{\boldsymbol\gamma}$ denote the corresponding hatted charge data, and let $\mathscr{Z}_{n-1}$ index the $n-1$ possible adjacent combinations. Define

$$\mathscr{G}_n(\hat{\boldsymbol\gamma};\tau_2,\beta)=\hat\Phi^M_{n-1}(\{\boldsymbol v_{\ell}\};\sqrt{2\tau_2}(\boldsymbol q+\beta\boldsymbol\theta),\sqrt{2\tau_2}\,\beta\boldsymbol\theta).$$

In the one-modulus case, the functions $\mathscr{R}^{\rm ref}_n$ determining the coefficients are given by

$$\mathscr{R}^{\rm ref}_n(\hat{\boldsymbol\gamma};\tau_2,\beta)=\frac{1}{2^{n-1}}\sum_{\mathcal J\subseteq\mathscr{Z}_{n-1}}b_{|\mathcal J|}\,\delta_{\mathcal J}\,\mathscr{G}_{n-|\mathcal J|}(\hat{\boldsymbol\gamma}_{\mathcal J};\tau_2,\beta),$$

where $b_n$ are the Taylor coefficients of $\tanh(x)/x$, $b_{n-2}=\frac{2^n(2^n-1)}{n!}B_n$, $\delta_{\mathcal J}=\prod_{k\in\mathcal J}\delta_{\Gamma_k}$, and $\Gamma_k=\sum_{i=1}^k\sum_{j=k+1}^n\gamma_{ij}$. The datum $\hat{\boldsymbol\gamma}_{\mathcal J}$ is obtained by combining each $\hat\gamma_i$ with the next charge for $i\in\mathcal J$. One-modulus refined-coefficient conjecture. In the one-modulus case, the functions $\mathscr{R}^{\rm ref}_n$ are given by the displayed finite sum with the coefficients and Kronecker-delta factors specified above. The observation motivating this formula is that, for $n=3$ and $4$, the sum over Schröder trees undergoes a large cancellation, leaving only this type of term. The claim is supported in those cases by the stated cancellation, but its general validity is not established in the supplied text.