Square Hamilton cycle conjecture for graphs with sublinear independence number

Let $\varepsilon>0$. A graph $G$ has independence number $\alpha(G)$ equal to the maximum size of a set of pairwise nonadjacent vertices, and the square of a Hamilton cycle is the graph obtained from a Hamilton cycle by joining every pair of vertices at distance at most two on the cycle. Square Hamilton cycle conjecture. There exist $\gamma>0$ and $n_0\in\mathbb{N}$ such that, for every $n$-vertex graph $G$ with $n\geq n_0$, if

$$\delta(G)\geq (1/2+\varepsilon)n \quad\text{and}\quad \alpha(G)\leq \gamma n,$$

then $G$ contains the square of a Hamilton cycle. This would strengthen the known triangle-factor result for graphs with the same minimum-degree and independence-number conditions, and is posed as a natural next step toward a bandwidth theorem for graphs with sublinear independence number.