New bounds on the Graham-Pollak theorem for hypergraphs

For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-uniform hypergraphs whose edge sets partition the complete $r$-uniform hypergraph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. Let $c_r$ be the smallest constant such that $f_r(n)\le c_r(1+o(