Häggkvist's transversal cycle-factor conjecture for blow-ups of cycles

Let $C_k$ be the cycle on $k$ vertices, and let the $n$-blow-up of $C_k$ be the graph with vertex parts $V_1,\dots,V_k$, each of size $n$, where consecutive parts are joined according to the edges of $C_k$. For a spanning subgraph $G$ of this blow-up, write $\delta^*(G)$ for its minimum degree between consecutive vertex parts. A transversal $C_k$-factor is a collection of $n$ vertex-disjoint copies of $C_k$, each containing one vertex from every part.

Häggkvist's conjecture. For every $k\geq 3$, if

$$\delta^*(G)\geq \left(1+\frac{1}{k}\right)\frac{n}{2}+1,$$

then $G$ has a transversal $C_k$-factor.

The conjecture is tight when $k=3$, while the example described in the source shows that for $k\geq4$ the degree threshold cannot be decreased by more than $1$.