Pósa's conjecture on the square of a Hamiltonian cycle

Let $G$ be a graph on $n$ vertices. The square of a Hamiltonian cycle is $C_n^2$, where vertices at cyclic distance at most two are adjacent. Pósa's conjecture. If

$$\delta(G)\geq\frac{2n}{3},$$

then $C_n^2\subseteq G$. This conjecture would significantly strengthen the Corrádi–Hajnal theorem and implies the relevant degree-two case of the Bollobás–Eldridge–Catlin conjecture.