TITLE: Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?
QUESTION [9 upvotes]: In Montgomery's A Tour of Subriemannian Geometries, Their Geodesics and Applications, problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?". 
Can someone explain what are the major difficulties faced in proving the same ? Suggestions for suitable references are also welcome. Thanks.

REPLY [2 votes]: According to the MathSciNet, the first general contribution to the solution of the problem is given in the following paper:
E. Hakavuori, E. Le Donne,
Non-minimality of corners in subriemannian geometry.
Invent. Math. 206 (2016), no. 3, 693–704. 
(MathSciNet review).
The paper mentioned by Richard Montgomery is:
E. Le Donne, G. P. Leonardi, R. Monti, D. Vittone,
Extremal curves in nilpotent Lie groups.
Geom. Funct. Anal. 23 (2013), no. 4, 1371–1401. 
(MathSciNet review).