Morse-index transition conjecture for regular polygons

Let $N>4$, let $A$ be the exponent of the homogeneous potential, and let $f$ be the function whose Morse index is considered on the quotient configuration space $\mathcal{C}_N$. For each fixed $N$, write $A_N$ for a possible transition exponent. Morse-index transition conjecture. There exists a unique value $A_N$ for each $N>4$ such that, for $A<A_N$, the Morse index of $f$ for the regular polygon on $\mathcal{C}_N$ is $N-5$, while for $A>A_N$ it is $N-3$. Moreover, the values $A_N$ decrease monotonically with $N$ and satisfy

$$\lim_{N\rightarrow\infty}A_N=2.$$

This conjecture is motivated by numerical investigations and strengthens the preceding theorem; the existence, uniqueness, monotonicity, and limiting behavior of the transition values remain unproved.