Conjecture on the extremal obstruction to perfect matchings

Let $H$ be a 3-graph of order $n$, let $\sigma_2(H)$ denote the minimum $\deg(u)+\deg(v)$ over adjacent vertices $u,v$, and let $H^{1,2}_{n,x,y}$ be the construction defined in the source for parameters $x,y$. Extremal perfect-matching conjecture. There exists $n_0\in\mathbb{N}$ such that, for every 3-graph $H$ of order $n\ge n_0$ without isolated vertices, if

$$\sigma_2(H)>\max\left\{\sigma_2(H^{1,2}_{n,x,y}):0\le x,y\le\frac n3-1,\ x+y\le\frac n3-1\right\},$$

then $H$ contains no perfect matching if and only if $H$ is a subgraph of $H_{n,n/3}^2$. The source explains that an earlier conjecture fails when $s=n/3$ and near that range, motivating this strengthened extremal formulation; its resolution is not supplied.