Lopsided fractional Erdős matching conjecture for hypergraph tilings

Let $F$ be a $k$-graph, let $v(F)$ denote its number of vertices, and let $G_{n,i,\beta}$ be the space-barrier $k$-graph associated with parameters $n$, $i$, and $\beta$. Write $e(G_{n,i,\beta})$ for its number of edges, and let $\operatorname{\delta}_0(F;\beta)$ denote the corresponding fractional tiling threshold. For $0\leq\beta\leq 1/v(F)$, the conjecture asserts

Lopsided fractional Erdős matching conjecture. For every $k$-graph $F$ and $0\leq\beta\leq 1/v(F)$,

$$\operatorname{\delta}_0(F;\beta)=\max_{i\in[k]}\limsup_{n\to\infty}\frac{e(G_{n,i,\beta})}{\binom{n}{k}}.$$

This would identify the asymptotic fractional tiling threshold with the largest density produced by the space barriers. The paper presents it as an open direction motivated by the fractional Erdős matching conjecture, and no resolution is supplied here.