Six-body Morse-polynomial conjecture

Consider the equal-mass planar six-body problem, with exponent $A$, function $f$, and Morse polynomial $M(t)$; let $P(t)$ be the Poincaré polynomial of the quotient configuration space. Six-body Morse-polynomial conjecture. For sufficiently large $A$,

$$M(t)=384+1440t+2520t^2+2520t^3+1080t^4=P(t)+(1+t)(383+1043t+1406t^2+960t^3),$$

and for $A=3$,

$$M(t)=384+840t+1080t^2+960t^3+360t^4=P(t)+(1+t)(383+443t+566t^2+240t^3).$$

The claim is supported by numerical calculations and by independent fixed-point investigations, but the asserted Morse polynomials are not proved in the source.