Newtonian Morse-polynomial conjecture for seven-, eight-, and nine-body problems

Let $f$ be the function for the equal-mass planar $N$-body problem, and let $M(t)$ denote its Morse polynomial. Newtonian Morse-polynomial conjecture. For $A=3$, the Morse polynomials are

$$M(t)=120(7+84t+132t^2+105t^3+84t^4+35t^5)$$

for $N=7$,

$$M(t)=720(8+56t+224t^2+301t^3+210t^4+112t^5+28t^6)$$

for $N=8$, and

$$M(t)=5040(81+216t+384t^2+732t^3+746t^4+396t^5+168t^6+36t^7)$$

for $N=9$. These conjectured central-configuration counts and Morse indices agree with the numerical results cited in the source, subject to the noted possible typographical issue in the nine-body data; a rigorous completeness proof remains open.