The finite-rhombi 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. Finite-rhombi conjecture. For every integral curve $\gamma\in\mathcal M_n$, there is a finite set of unit rhombi $\rho_1,\ldots,\rho_k\in\mathcal M_4$ and a dome over $\gamma\cup\rho_1\cup\cdots\cup\rho_k$. The paper presents this as an independently interesting reduction toward the single-rhombus conjecture.