The tangled-cord extremal conjecture for Hamiltonian sets in simple assembly graphs

Let $G$) be a simple assembly graph with $n$ rigid vertices, each of degree $4$, and let $F_k$ denote the $k$th Fibonacci number. A Hamiltonian set of polygonal paths is a Hamiltonian set as defined for such assembly graphs. Tangled-cord extremal conjecture. The upper bound for the maximal number of Hamiltonian sets of polygonal paths is

$$F_{2n+1}-1.$$

This bound is achieved only by the tangled cord with $n$ rigid vertices of degree $4$. Earlier work established the upper bound and showed that it is tight; the conjecture concerns the uniqueness of the extremal graph, and the present paper develops equivalent combinatorial conditions intended to prove it.