In the paper 
*[The complexity of recognizing linear systems with certain integrality properties][1]*, Guoli Ding, Li Feng & Wenan Zang, Mathematical Programming **114**, pages 321–334 (2008)
the authors prove that testing integrality is coNP-complete.


  [1]: https://doi.org/10.1007/s10107-007-0103-y