Crazy Knight’s Tour Problem for toroidal partially filled arrays

Determine whether, for a given toroidal partially filled array, there exist row and column orientations R and C such that the alternating traversal list L_{R,C}(i_1,j_1) from an initial filled cell covers every filled cell of the array.

Background

The Crazy Knight’s Tour Problem is defined for a toroidal partially filled array by fixing orientations for every row and column and then repeatedly moving to the next filled cell in the current row and subsequently the next filled cell in the resulting column. The problem asks whether suitable orientations can make this traversal visit all filled positions.

This question is relevant to the paper’s biembedding results because a solution supplies compatible orderings for globally simple relative Heffter arrays. The paper invokes previously known sufficient and necessary conditions for particular classes of arrays, but does not resolve the problem for arbitrary toroidal partially filled arrays.