The asymptotic $B_k$-free hypergraph bound

Fix $k\geq 2$, and let $H$ be a $3$-uniform $B_k$-free hypergraph on $n$ vertices, where $n$ is sufficiently large. Write $e(H)$ for the number of hyperedges of $H$.

The $B_k$-free hypergraph conjecture. One should have

$$e(H)\leq \dfrac{n^2}{8}.$$

This is the hypergraph analogue of the graph bounds for triangle-free and book-free graphs. The paper proves the asymptotic upper bound $\operatorname{ex}_3(n,B_k)=\frac{n^2}{8}(1+o(1))$, while the exact bound stated here remains open in the source.