Baugher's integral identity for the Morse-index coefficient

Let $m$ be the complex dimension, let $m\leq q\leq 2m$, and let $Y_{2m-q}$, $\Delta(\lambda)$, $\Delta(\xi)$ and the integration variables be as in the source's preceding definitions. Baugher's conjecture. The coefficient $\beta_{2q}(m)$ is given by

$$\beta_{2q}(m)=\frac{(-i)^{m(m-1)/2}}{4\pi^{2m}\prod_{j=1}^{m-1}j!}\int_{Y_{2m-q}}\int_{\mathbb R}\cdots\int_{\mathbb R}\Delta(\lambda)\Delta(\xi)\prod_{j=1}^m|\lambda_j|e^{i\langle\lambda,\xi\rangle}\mathcal{J}(\xi)\,d\xi_1\cdots d\xi_m\,d\lambda,$$

where

$$\mathcal{J}(\xi)=\frac{4}{(m+1)(m+2)(m+3)\left(1-\frac{i}{2}\sum_j\xi_j\right)^2\prod_{j\leq k}\left[1+\frac{i}{2}(\xi_j+\xi_k)\right]}.$$

The identity was conjectured from symbolic computations and is intended to imply positivity of $\beta_{2q}(m)$; its general validity remains open.