  We exhibit conjugate points on the Stiefel manifold endowed with any member
of the family of Riemannian metrics introduced by H\"uper et al. (2021). This
family contains the well-known canonical and Euclidean metrics. An upper bound
on the injectivity radius of the Stiefel manifold in the considered metric is
then obtained as the minimum between the length of the geodesic along which the
points are conjugate and the length of certain geodesic loops. Numerical
experiments support the conjecture that the obtained upper bound is in fact
equal to the injectivity radius.
