Universal odd critical exponent conjecture for exponential-kernel minimizers

Consider the minimum-energy problem for $k$ points on the unit interval with pairwise kernel $e^{-|x-y|^q}$, and write an odd number of points as

$$k=2m+1.$$

Let the global critical exponent be the value of $q$ at which the globally minimizing configuration changes phase, and let the universal odd value be the value given by equation. Universal odd critical exponent conjecture. For every odd $k=2m+1$, the true global critical exponent equals the universal odd value in equation. The conjecture is motivated by the explicit branch crossing and numerical evidence for $k=3,5,7,9,11,13,15,19$; the unresolved issue is excluding a lower-energy competing branch with more than one interior point.