 Solution

In the usual exchange-hole representation, the hole around an electron integrates to charge deficit $-1$. Far outside a finite system, the localized hole therefore looks like a unit opposite charge at distance $r$. The Coulomb interaction has leading scale $-1/r$, and the exchange pair energy carries the usual factor $1/2$ to avoid double counting, giving exchange energy per electron $-1/(2r)$.

This energy-per-electron tail should not be confused with the exchange or exchange-correlation potential, whose leading coefficient is generally $-1/r$ in the corresponding exact finite-system setting. Local energy-density representations also require a specified gauge; the question uses the conventional exchange-hole interpretation.