NP completeness of Hamiltonian cycle for the family of *dual graphs* to plane, cubic, triply connected graphs?
The following paper shows that the Hamiltonian cycle problem is NP-complete in maximal planar graphs:


  A. Wigderson
  The Complexity of the Hamiltonian Circuit Problem for Maximal Planar Graphs
  Technical Report #298, Department of EECS, Princeton University, February 1982.
  https://www.math.ias.edu/avi/node/820


In a maximal planar graph, every face is a triangle; hence the dual of a  maximal planar graph is cubic. It is easy to see that the dual of a maximal planar graph is triply connected. 

Therefore your problem is NP-complete.