Five-body Morse-structure conjecture

Let $f$ be the function for the equal-mass planar five-body problem, let $\mathcal{C}_5$ be the quotient configuration space, and let $P(t)$ be its Poincaré polynomial. Five-body Morse-structure conjecture. There are unique values $A_5\in(6.755,6.756)$ and $A_c\in(7.5636,7.5638)$ such that

$$M(t)=54+120t+120t^2+60t^3=P(t)+(1+t)(53+58t+36t^2)$$

for $2\leq A<A_5$,

$$M(t)=150+240t+144t^2+60t^3=P(t)+(1+t)(149+82t+36t^2)$$

for $A_5<A<A_c$, and

$$M(t)=120+240t+174t^2+60t^3=P(t)+(1+t)(119+112t+36t^2)$$

for $A_c<A$. These numerical predictions summarize the proposed bifurcation and Morse-index structure of the five-body problem; their completeness and the asserted transition values remain unproved.