The conjectured fractional codegree threshold for generalized Steiner systems

For integers $n\geq r\geq 2$, a fractional $(r-1,r,n)$-Steiner system in an $r$-uniform hypergraph $H$ is a function $\varphi:E(H)\to\mathbb{R}_{\geq0}$ such that $\deg_H^\varphi(p)=1$ for every $(r-1)$-set $p\in V(H)^{(r-1)}$. Fractional Steiner-system threshold conjecture. The minimum codegree threshold for a fractional $(r-1,r,n)$-Steiner system is

$$\frac{(r-1)n}{r}+\Theta(1).$$

This generalizes the paper's parity-based lower-bound construction and asks for the asymptotically sharp fractional codegree threshold; it remains open.