Revised asymptotic tiling conjecture for rigid spanning hypergraphs

Let $r\ge 3$ and $1 \le s_1 \le \cdots \le s_r$ be integers, and let $m\coloneqq s_1+\cdots+s_r$. A spanning subgraph $F$ of $K_{s_1,\ldots,s_r}^{r}$ is rigid if, for some such integers with $v(F)=s_1+\cdots+s_r$, it satisfies $\tau_i(F)=s_1+\cdots+s_i$ for every $i\in [r-1]$, where $\tau_i(F)$ is the minimum size of a vertex set meeting every edge of $F$ in at least $i$ vertices. For integers $n\ge r$ and $1\le i\le r$, let $G_{n,i,\beta}(s_1,\ldots,s_r)$ be the $n$-vertex $r$-graph whose edges are the $r$-sets meeting a distinguished set of size $\lfloor\beta(s_1+\cdots+s_i)n\rfloor-1$ in at least $i$ vertices. Revised tiling conjecture. For every real number $\beta\in\left(0,\frac{1}{m}\right)$,

$$\mathrm{ex}(n,\beta n \cdot F)=\max\left\{|G_{n,i,\beta}(s_1,\ldots,s_r)|\colon i\in [r]\right\}+o(n^r).$$

The revision excludes the previously refuted formulation by imposing rigidity, a condition satisfied by complete multipartite hypergraphs and intended to capture the relevant covering-number obstructions. The source does not state a resolution.