Glazyrin–Pak cobordism conjecture for integral curves

Let $\gamma$ be a closed piecewise linear curve in three-dimensional Euclidean space $\mathbb{E}$ whose intervals all have unit length; such a curve is an integral curve. A unit rhombus is the boundary curve of a rhombus with unit-length sides, and a dome over a curve is a piecewise linear complex with that boundary whose facets are unit triangles. Two curves are cobordant if their union bounds such a dome.

Glazyrin–Pak cobordism conjecture. For every integral curve $\gamma$, there is a unit rhombus $\rho$ and a dome over $\gamma\cup\rho$.

This conjecture proposes that the obstruction found for unit rhombi is the only restriction preventing a general integral curve from being domed. The paper disproves the corresponding assertion for oriented domes, although it proves that every integral curve is orientably cobordant to a finite union of unit rhombi.