Conlon–Fox–Kwan–Sudakov surplus conjecture for hypergraph cuts

Prove that, for every pair of integers 2\leq r\leq k with either k\geq 4 or r\geq 3, every k-uniform hypergraph with m edges has an r-cut whose surplus over the expected size of a uniformly random r-cut is \Omega(m^{2/3}).

Background

For a k-uniform hypergraph, an r-cut is a partition of the vertex set into r parts, and its surplus is the cut size minus the expected cut size under a uniformly random r-partition. Conlon, Fox, Kwan, and Sudakov previously proved a general lower bound of \Omega(m^{5/9}) for the surplus, except in the graph and 3-uniform 2-cut exceptional cases.

They observed that random k-uniform hypergraphs with appropriate density have surplus O(m^{2/3}), suggesting that the exponent 2/3 is best possible. The paper establishes only an approximate form of this conjecture, namely an \Omega(m^{2/3-\varepsilon}) bound for sufficiently large uniformity k depending on \varepsilon; the exact \Omega(m^{2/3}) assertion therefore remains unresolved in the stated generality.