The cubic-map enumeration conjecture for bipartite chord diagrams

Let $\mathcal{C}_{n}(\\{T_{2k+1},B_{2k+1}\\}_{k\geqslant 1})$ be the set of connected chord diagrams of size $n$ avoiding all odd top and bottom cycles. The cubic-map enumeration conjecture. The cardinality of this set equals the cardinality of the set of all $3$-edge-connected rooted cubic maps with $2n$ vertices and a distinguished Hamiltonian cycle. The proposed correspondence connects bipartite chord diagrams with a natural class of cubic maps; the source gives no proof, so the conjecture remains open.