Parity conjecture for the triple twist of slab tilings

Let ${\cal R}$ be a region tileable by slabs. Let $\operatorname{Tw}_{\kappa_x}({\mathbf t})$, $\operatorname{Tw}_{\kappa_y}({\mathbf t})$, and $\operatorname{Tw}_{\kappa_z}({\mathbf t})$ denote the three triple-twist integers of a slab tiling ${\mathbf t}$ of ${\cal R}$, in the directions $x$, $y$, and $z$. Then $t_x,t_y,t_z\in\{0,1\}\subset\mathbb Z$ are constants associated with ${\cal R}$.

Triple-twist parity conjecture. For every slab tiling ${\mathbf t}$ of ${\cal R}$, the integers

$$\operatorname{Tw}_{\kappa_x}({\mathbf t})-t_x,\quad \operatorname{Tw}_{\kappa_y}({\mathbf t})-t_y,\quad \operatorname{Tw}_{\kappa_z}({\mathbf t})-t_z$$

are all even. If ${\cal R}$ is a box, then $t_x=t_y=t_z=0$.

This conjecture asserts that the parities of the three triple-twist coordinates are invariant across all slab tilings of a region. The paper presents it as its main conjecture; no resolution is supplied in the source.