Kalai’s conjecture for hypertrees

Prove Kalai’s conjecture that every r-uniform hypertree with t hyperedges satisfies ex_r(n,T^r) ≤ (t−1)\binom{n}{r−1}/r.

Background

An r-uniform hypertree is constructed by adding each hyperedge after the first so that it introduces one new vertex while all remaining vertices lie in an earlier hyperedge. The paper states Kalai’s conjectured upper bound for the Turán number of such hypertrees and notes that it is known only for various classes of hypertrees. The conjecture is relevant because suspensions of graph trees are themselves hypertrees.