Is the isometric path cover problem NP-complete?
This question is mentioned in a recent (2017) paper, Strong geodetic problem in networks:
computational complexity and solution for
Apollonian networks by Paul Manuel, Sandi Klavžar, Antony Xavier,
Andrew Arokiaraj, and Elizabeth Thomas. On page 8, they state:


  To our knowledge, the complexity status of the isometric path problem is
  not known.