Towards Pósa's Conjecture for $3$-graphs

We prove that every $3$-graph $H$ on $n$ vertices with minimum codegree ${\delta }_2(H)\ge 7n\mathrm{/}9+o(n)$ contains the square of a tight Hamilton cycle. This strengthens a theorem of Bedenknecht and Reiher that ${\delta }_2(H)\ge 4n\mathrm{/}5+o(n)$ is sufficient. The central novelty of our arguments is an improved understanding of the connectivity structure of $3$-graphs with large minimum codegree.