The conical kinematics amplitude conjecture

Let $\mathcal{K}(k,n)$ be the kinematic space, let $J\in\binom{[n]}{k}^{\mathrm{nf}}$, and define

$$\gamma_J=\sum_{\ell=1}^{k-1}\left(\alpha_{\ell,j_\ell-(\ell-1)}-\alpha_{\ell,j_{\ell+1}-(\ell-1)-1}\right),$$

where $(i,j)\in[1,k-1]\times[1,n-k+1]$. For $\alpha\in\mathbb{R}^{(k-1)(n-k+1)}$, choose the coordinate functions $s$ on $\mathcal{K}(k,n)$ by solving $\eta_J(s)=\gamma_J$. Conical kinematics amplitude conjecture. The scattering equations have a unique solution, and

$$m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)\big\vert_{\eta_J=\gamma_J}=\prod_{i=1}^{k-1}\left(\frac{\alpha_{i,1}-\alpha_{i,n-k+1}}{\prod_{j=1}^{n-k}(\alpha_{i,j}-\alpha_{i,j+1})}\right).$$

This conjectural formula gives an explicit evaluation of the amplitude for conical kinematics and is presented as a generalization of the $k=2$ construction; the supplied text gives no proof or resolution.