The quarter-range tripartite hypergraph degree-sequence conjecture

A tripartite hypergraph degree sequence on $n+n+n$ vertices assigns degrees to the three vertex classes; it is graphic if it has a tripartite hypergraph realization. The quarter-range tripartite hypergraph degree-sequence conjecture. For every $\varepsilon>0$, sufficiently large $n$, any such degree sequence with equal degree sums in the three vertex classes, every degree between $n^2/4$ and $3n^2/4$, and each common class degree sum either at most

$$\left(\frac{7-\sqrt{17}-\varepsilon}{8}\right)n^3$$

or at least

$$\left(\frac{1+\sqrt{17}+\varepsilon}{8}\right)n^3$$

is graphic. Conversely, whenever $0<\varepsilon<(\sqrt{17}-3)/4$, for all sufficiently large $n$ there exists a non-graphic tripartite hypergraph degree sequence with every degree in the same interval and common class degree sum

$$\left\lceil\left(\frac{5-\sqrt{17}}{4}+\varepsilon\right)\frac{3n^3}{4}+\left(\frac{\sqrt{17}-1}{4}-\varepsilon\right)\frac{n^3}{4}\right\rceil.$$

This conjecture refines the quarter-to-three-quarter degree-range observation by predicting exceptional degree-sum ranges where graphicness is guaranteed and a complementary range containing non-graphic sequences.