The rhombus reduction conjecture for integral curves

Let $\mathcal M_n$ denote the space of integral curves with $n$ edges, and let $\mathcal M_4$ denote the corresponding space of unit rhombi. Rhombus reduction conjecture. For every integral curve $\gamma\in\mathcal M_n$, there is a unit rhombus $\rho\in\mathcal M_4$ and a dome over $\gamma\cup\rho$. This is described as a cobordism principle for domes: whether an integral curve can be domed should reduce to adjoining a single rhombus.