Rainbow Hamiltonian cycle conjecture for graph families satisfying an Ore-type condition

Let $n\in\mathbb{N}$ with $n\geq 3$, and let $\mathcal{G}=\{G_i:i\in[n]\}$ be a family of $n$-vertex graphs with the same vertex set; the graphs in the family may be identical. For a graph $G$, let $\sigma(G)$ denote the minimum, over all nonadjacent vertex pairs $u,v$, of $d_G(u)+d_G(v)$. Rainbow Hamiltonian cycle conjecture. If $\sigma(G_i)\geq n$ for every $i\in[n]$, then $\mathcal{G}$ contains a rainbow Hamiltonian cycle. This extends Ore's Hamiltonicity condition from a single graph to a family of graphs; the supplied text gives no resolution, so the conjecture is recorded as open.