General minimum codegree conjecture for perfect matchings in uniform hypergraphs

Suppose that $k,\ell\in\mathbb N$ with $\ell\leq k-1$. For a $k$-uniform hypergraph $H$, let $\delta_\ell(H)$ denote its minimum $\ell$-degree, and let $m_\ell(k,n)$ be the smallest integer $m$ such that every $k$-uniform hypergraph on $n$ vertices with $\delta_\ell(H)\geq m$ contains a perfect matching. Let $\delta(n,k,\ell)$ be the maximum minimum $\ell$-degree among the divisibility-barrier hypergraphs described in the source. Then the general minimum codegree conjecture asserts that, for all sufficiently large $n\in k\mathbb N$,

$$m_\ell(k,n)=\max\left\{\delta(n,k,\ell),\ \binom{n-\ell}{k-\ell}-\binom{(1-1/k)n-\ell+1}{k-\ell}\right\}+1.$$

This conjecture combines the divisibility and space-barrier constructions, each of which gives a lower bound for the minimum $\ell$-degree forcing a perfect matching. It is known in several cases, including $\ell\geq k/2$ and various small pairs $(k,\ell)$, but remains open in general.