What is the computational complexity of solving an instance of this problem?
The posed problem has no free cells, and in fact the labels are all in a rectangular board R. This seems the most interesting specific variant, for it is left possible in [BBD+07] that, if there is a solution for R, it is ``uniquely rollable.'' They establish that there are boards with labeled and blocked (i.e., forbidden) cells for which rollable Hamiltonian cycles are not unique, but they leave open fully labeled boards. The uniquely-rollable conjecture is settled for all boards with side lengths at most 8.