I am currently interested in hamilton cycles (i.e. a cycle through every vertex) in planar triangulations (i.e. planar graphs with every face a triangle).
There are non-hamiltonian planar triangulations about which I have nothing to say or ask. However if a planar triangulation is hamiltonian then it has at least 4 hamilton cycles, and Hakimi, Schmeichel and Thomassen (HST) found an infinite family of planar triangulations with exactly 4 hamilton cycles. However the only infinite families of planar triangulations with a bounded number of hamilton cycles have separating triangles (i.e. 3-vertex clique-cutsets) and can in some sense be viewed as "reducible'' configurations.
Therefore HST asked what is the smallest number of hamilton cycles that a $4$-connected planar triangulation on $n$ vertices can have, and conjectured that the answer is $2(n-2)(n-4)$ hamilton cycles. Furthermore they conjectured that the unique example realising this is the complete join of $2K_1$ (two isolated vertices) to a cycle of length $n-2$.
My question is whether this has been proved or any progress has been made on it. I have searched fairly hard through MathSciNet and Google, and have convinced myself that there are no small counterexamples.