Asymptotic equivalence of weak saturation and rho-saturation

Let $r\ge 1$ and $s\ge 1$ be integers, and let $H$ be an $r$-uniform hypergraph with $s(H)=s$. Asymptotic equivalence conjecture. Then

$$\operatorname{\mathrm{wsat}}(n,H)=(1+o(1))\operatorname{\rho\mathrm{-sat}}(n,H).$$

The quantity $\operatorname{\rho\mathrm{-sat}}(n,H)$ is defined as the best lower bound on weak saturation obtainable from the paper's polymatroid lower-bound theorem. The conjecture would identify its asymptotic coefficient with that of weak saturation, beyond the already established equality of their orders of growth.