The asymptotics of the Kohn-Sham (KS) exact exchange potential Vx(z ) of a jelliumlike semi-infinite metal is investigated in the framework of the optimized-effective-potential formalism of density-functional theory. Our numerical calculations clearly show that deep into the vacuum side of the surface Vx(z ) ∝e2ln(a z ) /z with a being a system-dependent constant, thus, confirming the analytical calculations reported in Phys. Rev. B 81, 121106(R) (2010), 10.1103/PhysRevB.81.121106. A criticism of this work published in Phys. Rev. B 85, 115124 (2012), 10.1103/PhysRevB.85.115124 is shown to be incorrect. Our rigorous exchange-only results provide strong constraints both for the building of approximate exchange functionals and for the determination of the still unknown KS correlation potential.