Real-root and sign-distribution conjecture for characteristic polynomials in hyperbolic space

Let $A_1,\ldots,A_m$ be $m\geq n+2$ points in $H^n$, and let $\alpha_1,\ldots,\alpha_m$ satisfy $\sum_{i=1}^m\alpha_iA_i=0$. For $0\leq k\leq n$, let $c_k$ be the constants defining the characteristic polynomial

$$f(x)=c_{0}x^{n}-c_{1}x^{n-1}+\cdots+(-1)^{i}c_{i}x^{n-i}+\cdots+(-1)^{n}c_{n}.$$

Characteristic-polynomial conjecture. All roots of $f(x)$ are real numbers. If $m=n+2$, the points are in general position in $H^n$, and exactly $s$ of the $\alpha_i$ are positive, then $f(x)$ has $s-1$ positive and $n+1-s$ negative roots.

This gives both a general real-root assertion and a more precise root-sign count in the minimally dependent, general-position case. The statement concerns the hyperbolic analogue of the characteristic-polynomial results established for Euclidean and spherical configurations; its resolution is not supplied here.