Asymptotic Turán-density upper-bound conjecture for suspensions

For $s\geq2$, let $K_s^{(2)}$ be the complete $2$-graph on $s$ vertices, let $S(K_s^{(2)})$ be its suspension, and let $4\pi(S(K_s^{(2)}))4$ denote its Turán density. Asymptotic suspension upper-bound conjecture. For every $4\varepsilon>04$ there exists $4s_0=s_0(\varepsilon)4$ such that for all $4s\geq s_04$,

$$\pi(S(K_s^{(2)}))\leq 1-\frac{2-\varepsilon}{s}.$$

The paper gives the general upper bound $4\pi(S(K_s^{(2)}))\leq1-1/(s-1)4$ and a construction with asymptotic lower bound $1-1/(2s)+O(1/s^2)$. Determining the constant in the first-order asymptotics is left open.