Odd-cycle-different Hamiltonian paths conjecture

For integers $k>1$, let $C_{2k+1}$ denote the cycle of odd length $2k+1$. Two Hamiltonian paths on $n$ vertices are $C_{2k+1}$-different if their union contains a subgraph isomorphic to $C_{2k+1}$. Let the number of balanced bipartitions of an $n$-element ground set be the corresponding extremal benchmark.

Odd-cycle-different Hamiltonian paths conjecture. If $k>1$ and $n$ is large enough, the maximal number of pairwise $C_{2k+1}$-different Hamiltonian paths on $n$ vertices is equal to the number of balanced bipartitions of the ground set $n$.

This is proposed as a relaxed extension of the paper's theorem for triangles. The source presents it as open; it also gives supporting asymptotic results in some cases.