General asymptotic and stability conjecture for tree blowups

Let $\mathcal{T}\in\mathcal{T}_{s,t}(a,b)$, where $b\leq a<r-1$, and set $\sigma=\sigma(\mathcal{T})=\min\{s,t\}$. Let $H$ be a $\mathcal{T}$-free $n$-vertex $r$-graph. General tree-blowup conjecture. For sufficiently large $n$,

$$|H|\leq (\sigma-1)\binom{n}{r-1}+o(n^{r-1}),$$

with equality only if $H$ is isomorphic to a hypergraph obtained from $\Psi_{\sigma-1}(n,r)$ by adding or deleting $o(n^{r-1})$ edges.

The paper states that its results determine the asymptotic behavior when $b\leq a<r$ in the established cases, while this conjecture covers the remaining general range except $a=r-1$; its resolution is not supplied.