Asymptotic equivalence of the edge functions for uniform systems

Let $r$ and $t$ be integers with $r\geq 3$ and $t\geq r$. For an $(r,t)$-system $(H,\mathcal{F})$, let $e_{r,t}(s)$ be the minimum of $e(H)$ over systems with $|\mathcal{F}|=s$, and let $e_{r,t}'(s)$ be the corresponding minimum over maximal systems. The notation $f(s)\asymp g(s)$ means that the two functions have the same order of magnitude. Asymptotic edge-function conjecture. For all integers $r\geq 3$ and $t\geq r$,

$$e_{r,t}(s)\asymp e_{r,t}'(s).$$

This compares arbitrary and maximal $(r,t)$-systems and predicts that imposing maximality does not change the asymptotic edge requirement.