The Cayley–Menger factorization conjecture for the radial metric determinant

Let $M=\frac{n(n-1)}{2}$ and let $g^{\mu\nu}$ be the contravariant metric matrix formed by the second-order coefficients of the radial operator $\Delta_{\rm rad}$ in the variables $\rho_{ij}$. Define

$$c_n(m)=c_1c_2\cdots c_{n-1}\,\frac{m_1+m_2+\cdots+m_n}{(m_1m_2\cdots m_n)^2},\qquad c_k=2^k(k!)^2.$$

Let $\widetilde V_n^2$ be the squared volume of the interaction polytope, and let $\widetilde V_{n-k}^2$ denote the corresponding weighted sums of squared face volumes, with $\widetilde V_1^2=1$ and $\widetilde V_n^2=V_n^2$.

Cayley–Menger factorization conjecture. The determinant $\det g^{\mu\nu}$ is homogeneous of degree $M$ in the $M$ variables $\rho_{ij}$ and factors as

$$\det g^{\mu\nu}=c_n(m)F_1F_2,$$

where $F_1=\widetilde V_n^2$ has degree $n-1$, while $F_2$ has degree $\frac{(n-1)(n-2)}{2}$, is a polynomial in the weighted face-volume variables, depends effectively on $n-1$ such polynomial variables, and satisfies $F_2\geq0$.

The factorization was checked for $n=2,3,4$ with arbitrary masses and for $n=5,6$ with equal masses, but the general claim remains unresolved in the supplied text.