Conjecture on the codegree threshold for spanning loose-cycle factors

There exists an integer $n_0$ such that for all $n\geq n_0$, the following holds. Let $\mathcal{C}$ be a $3$-graph consisting of vertex-disjoint loose cycles $C_{n_1},C_{n_2},\ldots,C_{n_r}$ such that

$$\sum_{i=1}^{r}n_i=n.$$

Let $k$ denote the number of loose cycles with odd lengths, and let $\delta_2(\mathcal{H})$ denote the minimum codegree of the $3$-graph $\mathcal{H}$. The spanning loose-cycle-factor conjecture. If $\mathcal{H}$ is an $n$-vertex $3$-graph with

$$\delta_2(\mathcal{H})\geq\frac{n+2k}{4},$$

then $\mathcal{H}$ contains $\mathcal{C}$ as a spanning subhypergraph. This conjecture proposes an asymptotically tight codegree threshold for embedding disjoint loose cycles in $3$-graphs. The paper establishes an asymptotic bound motivating the conjecture, while the stated exact threshold remains open.