Conjectured threshold for powers of Hamilton cycles

Let $\varrho(K_r)$ denote the graph parameter used in the source, and let $G$ be an $n$-vertex graph with minimum degree $\delta(G)$ and independence number $\alpha(G)$. The $r$-th power of a Hamilton cycle is obtained by joining every pair of vertices at cyclic distance at most $r$ on a Hamilton cycle.

Connecting-barrier conjecture. Given $\mu>0$ and $r\geq 4$, there exists $\alpha>0$ such that the following holds for sufficiently large $n$. If

$$\delta(G)\geq \frac{2-\varrho(K_r)}{3-2\varrho(K_r)}n+\mu n$$

and

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

then $G$ contains an $r$-th power of a Hamilton cycle.

The conjecture is motivated by a construction called the connecting barrier, which gives a matching asymptotic lower bound in the source for $r\geq 3$. Its resolution status is not specified here.