Degenerate Turán densities of sparse hypergraphs II: a solution to the Brown-Erdős-Sós problem for every uniformity

For fixed integers $r\ge 3,e\ge 3$, and $v\ge r+1$, let $f_r(n,v,e)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph in which the union of arbitrary $e$ distinct edges contains at least $v+1$ vertices. In 1973, Brown, Erdős and Sós proved that $f_r(n,er-(e-1)k,e)=\mathrm{\Theta }(n^k)$ and conjectured that the limit ${\lim }_{n\to \mathrm{\infty }}\frac{f_3(n,e+2,e)}{n^2}$ always exists for all fixed integers $e\ge 3$. In 2020 Shangguan and Tamo conjectured that the limit ${\lim }_{n\to \mathrm{\infty }}\frac{f_r(n,er-(e-1)k,e)}{n^k}$ always exists for all fixed integers $r>k\ge 2$ and $e\ge 3$, which contains the BES conjecture as a special case for $r=3,k=2$. Recently, based on a result of Glock, Joos, Kim, Kühn, Lichev, and Pikhurko, Delcourt and Postle proved the BES conjecture. Extending their result, we show that the limit ${\lim }_{n\to \mathrm{\infty }}\frac{f_r(n,er-2(e-1),e)}{n^2}$ always exists, thereby proving the BES conjecture for every uniformity.