Barnette's conjecture states that every cubic planar bipartite 3-connected graph admits Hamiltonian cycles.
Kelman claims that this conjecture is equivalent to a stronger one, which imposes some restrictions on the Hamiltonian cycles. The statement of this equivalence lies in his paper, "Constructions of cubic bipartite 3-connected graphs without Hamiltonian cycles", AMS Translations, Series 2 158, pp. 127–140.
It is Theorem 17 in this paper.
Unfortunately, Kelmans does not even give a sketch of the proof. I have read in detail the paper, but was not able to reconstruct the proof.
So I would like to know if :
1) these details are written somewhere.
2) is it true that if a Barnette graph is Hamiltonian, then Kelman's conditions can be imposed on that very graph (without any more assumption).