Asymptotic sharpness of the design lower bound for sunflower suspensions

Prove that the design-based lower bound for the Turán number of the sunflower S_rM_t is asymptotically sharp, namely that the corresponding construction gives the asymptotic value of ex_r(n,S_rM_t).

Background

The suspension S_rM_t consists of t r-edges sharing the same r−2 vertices, with the remaining pairs forming a matching; it is a sunflower hypergraph. A construction derived from an (n,r+2t−3,r−2,1)-design supplies a lower bound of order n^{r−2}. The paper cites a conjecture that this lower bound is asymptotically sharp, while noting that later work determines the order of magnitude for r≥4 but not necessarily the exact asymptotic constant.