Existence conjectures for solutions in small cuboids

Let $C(k,l,m)$ be the cuboid consisting of $klm$ unit cubes, and call a configuration a solution when it has the dismantling property used throughout the paper. Small-cuboid existence conjecture. There is a solution for $C(2,2,k)$ if and only if $k=5$; there is a solution for $C(2,3,k)$ if and only if $k=6$; and there is a solution for $C(3,3,k)$ if and only if $k$ is odd or $k=4$. These questions concern solutions in cuboids beyond the cases covered by the paper's general results. The supplied status evidence identifies this as part of the open questions about sequences of neighbouring-vertex moves forming a path.