The odd-uniform tight-cycle tiling threshold formula

Let $(k,s)$ be an admissible pair with $k\ge 3$ odd and $s\ge 5k^2$, and let $t_{k-1}(n,C_s)$ denote the minimum $(k-1)$-degree forcing a perfect $C_s$-tiling in a $k$-uniform hypergraph on $n$ vertices. Odd-uniform tiling threshold conjecture. Then

$$t_{k-1}(n,C_s) = (1/2 + k/(4s(k-1)+2k) + o(1))n.$$

This is the explicit asymptotic form of the lower-bound-tightness claim for odd $k$. The corresponding statement is open in the paper, while the analogous even-$k$ threshold is given as known.