Summary: \textit{O. Kowalski} and \textit{J. Szenthe} [Geom. Dedicata 81, No. 1-3, 209--214 (2000; Zbl 0980.53061), Erratum ibid. 84, 331--332 (2001)] proved that each homogeneous Riemannian manifold \((M, g)\) admits at least one homogeneous geodesic, i.e., a geodesic which is an orbit of a one-parameter group of isometries. In the present article we show that, for each dimension \(n\geq 4\), there is an \(n\)-dimensional (solvable) Lie group with a left-invariant metric which admits exactly one homogeneous geodesic through each point, up to a parametrization. Hence the result from O. Kowalski and J. Szenthe cannot be improved, in general.