Recently Hiai-Petz \cite{H} introduced two types of interesting geometries of positive-definite matrices whose geodesics are paths of operator means and then the author \cite{F1} showed these geometries have Finsler structures for all unitarily invariant norms. Though the geodesic is of the shortest length between fixed two matrices, the shortest paths are not unique in general as pointed out in \cite{H}. In this paper, we show that their geodesic is the unique shortest path in each Hiai-Petz geometry for all strongly convex unitarily invariant norms. As counter examples, we show that this uniqueness is false for Ky Fan norms.