Lang's asymptotic tiling conjecture for spanning subgraphs of complete multipartite hypergraphs

Let $r\ge 2$ and $1 \le s_1 \le \cdots \le s_r$ be integers, and let $m \coloneqq s_1 + \cdots + s_r$. Suppose that $F$ is a spanning subgraph of $K_{s_1, \ldots, s_r}^{r}$. 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 vertex set has a partition $V=V_1\cup V_2$ with $|V_1|=\lfloor\beta(s_1+\cdots+s_i)n\rfloor-1$ and whose edges are the $r$-sets meeting $V_1$ in at least $i$ vertices. Lang's 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).

This conjecture extends the Erdős Matching Conjecture from matchings to general spanning subgraphs of complete multipartite uniform hypergraphs; the source gives no resolution, and it is subsequently refuted by the revised conjecture's counterexamples.