Kamčev–Liebenau–Wormald conjecture on typical degree sequences of uniform hypergraphs

Let $2\le r\le n-2$ and let $\boldsymbol{d}$ denote a degree sequence of an $r$-uniform hypergraph with $m$ edges. Write $\mathcal{D}_r(n,m)$ for the degree-sequence distribution of a uniformly random $r$-uniform hypergraph with $m$ edges, and $\mathcal{T}_r(n,m)$ for the corresponding comparison distribution. Assume

$$\min\mathopen{}\mathclose\bgroup\originalleft\{m,\binom{n}{r}-m\aftergroup\egroup\originalright\}=\omega(\log n).$$

Kamčev–Liebenau–Wormald conjecture. There exists a set $\mathfrak{W}$ having probability $1-O(n^{-\omega(1)})$ in both $\mathcal{D}_r(n,m)$ and $\mathcal{T}_r(n,m)$ such that, uniformly for all $\boldsymbol{d}\in\mathfrak{W}$,

$$\mathbb{P}_{\mathcal{D}_r(n,m)}(\boldsymbol{d})=\mathbb{P}_{\mathcal{T}_r(n,m)}(\boldsymbol{d})\,(1+o(1)).$$

The conjecture asserts that the two degree-sequence distributions agree asymptotically for almost every degree sequence throughout the stated range. The paper proves this comparison under substantially stronger density and regularity hypotheses, leaving the full range of the conjecture as the unresolved motivation.