Exact local spectral thresholds for perfect matchings in $3$-graphs and $3$-partite $3$-graphs

For a $3$-uniform hypergraph $H$, let $σ(H)$ be the minimum over the vertices of $H$ of the spectral radius of the link. Lin, Lu, Yuan and Zhao conjectured that $σ(H)>\tfrac{2n}3-2$ forces a perfect matching in a $3$-graph of order $n$ divisible by three, and Lu and Yuan conjectured that $σ(H)>τ(q)$