Staden–Treglown conjecture for squares of Hamilton cycles

Let $G$ be an $n$-vertex graph, with minimum degree $\delta(G)$ and independence number $\alpha(G)$. A square of a Hamilton cycle is the graph obtained by joining vertices whose distance on the Hamilton cycle is at most two.

Staden–Treglown conjecture. For every $\mu>0$, there exist $\alpha>0$ and $_0\in \mathbb{N}$ such that the following holds. For every $n$-vertex graph $G$ with $\geq n_0$, if

$$\delta(G)\geq \left(\frac{1}{2}+\mu\right)n$$

and

$$\alpha(G)\leq \alpha n,$$

then $G$ contains a square of a Hamilton cycle.

This conjecture concerns Hamiltonian spanning structures in graphs with sublinear independence number. The source attributes it to Staden and Treglown; its resolution status is not specified here.