Four-body Morse-polynomial conjecture

Let $f$ be the function whose critical points represent central configurations in the four-body problem, and let $\mathcal{C}_4$ be the quotient configuration space with Poincaré polynomial $P$. Four-body Morse-polynomial conjecture. For $A>2$, the Morse polynomial of $f$ is

$$M=P+(1+t)(5+14t)=6+24t+20t^2.$$

The claim is a slightly stronger version of an earlier speculation and is supported by numerical analysis, but the authors state that they cannot prove it rigorously.