The near-quarter hypergraph degree-sequence conjecture

A hypergraph degree sequence on $n$ vertices is a sequence of vertex degrees, and it is graphic if it is realized by a $3$-uniform hypergraph; its degree sum is the sum of all vertex degrees. The near-quarter hypergraph degree-sequence conjecture. There exist $\varepsilon>0$ and $n_0\in\mathbb N$ such that, for every $n\ge n_0$, every hypergraph degree sequence on $n$ vertices whose degrees all lie between $\left(\frac14-\varepsilon\right)n^2$ and $\left(\frac14+\varepsilon\right)n^2$ and whose degree sum is divisible by $3$ is graphic. This proposes a uniform realizability result for dense $3$-uniform hypergraph degree sequences concentrated near one quarter of the maximum degree scale; the paper presents it as a weaker conjecture because the stronger possible sharp threshold is unclear.