The minimum codegree conjecture for perfect matchings in $K_4^3$-free 3-graphs

Let $H$ be an $n$-vertex $3$-uniform hypergraph, where $n$ is sufficiently large and divisible by $3$. Write $\delta_2(H)$ for its minimum codegree, and let $K_4^3$ denote the complete $3$-uniform hypergraph on four vertices. A perfect matching is a collection of vertex-disjoint edges covering all vertices of $H$.

$K_4^3$-free matching conjecture. For any $\gamma>0$, for sufficiently large $n\in 3\mathbb{N}$, if $H$ contains no copy of $K_4^3$ and

$$\delta_2(H)\geq \left(\frac{1}{3}+\gamma\right)n,$$

then $H$ contains a perfect matching.

The examples preceding the conjecture show that the constant $1/3$ is a natural lower-bound barrier for the minimum codegree. The supplied source gives no information about whether the conjecture has been resolved.