The short-range scattering exponent conjecture for time-dependent density functional theory

Consider the time-dependent density-matrix equation with self-interaction satisfying condition (SI2), and let $d$ denote the spatial dimension and $\beta$ the exponent in that condition. Short-range scattering exponent conjecture. The range of exponents $\beta$ for which short-range scattering holds is

$$\beta>1/d.$$

This is sharp in dimensions $1$ and $2$ according to the results discussed in the source, but is not optimal for $d=3$ and higher dimensions.