  H. Weyl in 1921 demonstrated that for a connected manifold of dimension
greater than $1$, if two Riemannian metrics are conformal and have the same
geodesics up to a reparametrization, then one metric is a constant scaling of
the other one. In the present paper, we investigate the analogous property for
sub-Riemannian metrics. In particular, we prove that the analogous statement,
called the Weyl projective rigidity, holds either in real analytic category for
all sub-Riemannian metrics on distributions with a specific property of their
complex abnormal extremals, called minimal order, or in smooth category for all
distributions such that all complex abnormal extremals of their nilpotent
approximations are of minimal order. This also shows, in real analytic
category, the genericity of distributions for which all sub-Riemannian metrics
are Weyl projectively rigid and genericity of Weyl projectively rigid
sub-Riemannian metrics on a given bracket generating distributions. Finally,
this allows us to get analogous genericity results for projective rigidity of
sub-Riemannian metrics, i.e.when the only sub-Riemannian metric having the same
sub-Riemannian geodesics, up to a reparametrization, with a given one, is a
constant scaling of this given one. This is the improvement of our results on
the genericity of weaker rigidity properties proved in recent paper
arXiv:1801.04257[math.DG].
