Extend sparse hypergraph enumeration asymptotics to larger maximum degrees

Determine whether the asymptotic enumeration formula of Aldesari and Greenhill for t-uniform hypergraphs with a prescribed degree sequence and forbidden edges extends to degree sequences whose maximum degree is o(n^{(t-1)/t}).

Background

The paper compares its sparse hypergraphicality results with an enumeration formula of Aldesari and Greenhill. That formula applies when the maximum degree d and total degree sum Sigma satisfy t^4 d^3=o(Sigma), a condition that requires d=o(n^{1/2}) in the relevant setting. The authors explicitly ask whether the enumeration result can be extended to the substantially broader maximum-degree range o(n^{(t-1)/t}).