Asymptotic 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 $C_{2k+1}$.

Asymptotic odd-cycle-different Hamiltonian paths conjecture. If $k>1$, then the maximal number of pairwise $C_{2k+1}$-different Hamiltonian paths on $n$ vertices is at least

$$2^{n-o(n)}.$$

This is stated as a weaker version of the preceding odd-cycle conjecture. The authors report that it is proved when $2k$ is a power of two, while the general case remains open.