Hypergraphs not containing a tight tree with a bounded trunk ~II: 3-trees with a trunk of size 2

A tight $r$-tree $T$ is an $r$-uniform hypergraph that has an edge-ordering $e_1, e_2, \dots, e_t$ such that for each $i\geq 2$, $e_i$ has a vertex $v_i$ that does not belong to any previous edge and $e_i-v_i$ is contained in $e_j$ for some $j<i$. Kalai conjectured in 1984 that every $n$-vertex $r$-uniform hypergraph with more than $\frac{t-1}{r}\binom{n}{r-1}$ edges contains every tight $r$-tree $T$ with $t$ edges. A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree $T'$ of $T$ such that vertices in $V(T)\setminus V(T')$ are leaves in $T$. Kalai's Conjecture was proved in 1987 for tight $r$-trees that have a trunk of size one. In a previous paper we proved an asymptotic version of Kalai's Conjecture for all tight $r$-trees that have a trunk of bounded size. In this paper we continue that work to establish the exact form of Kalai's Conjecture for all tight $3$-trees with at least $20$ edges that have a trunk of size two.