Conditional convergence after a finite excursion chain

Let $T:\mathbb N\to\mathbb N$ be the shortcut Collatz map: $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$. Write $T^k$ for its $k$-fold iterate, with $T^0(n)=n$. Write $\mathcal C(n)$ for the assertion that $T^j(n)=1$ for some $j\in\mathbb N$. Let $S$ be a finite list of segments $(k_i,A_i,D_i)$ of natural numbers, with $D_i>0$. Starting from $n$, define successive endpoints by applying $T^{k_i}$. Assume each segment satisfies $D_i(\text{endpoint}+3)\le A_i(\text{start}+3)$. Write $K=\sum_i k_i$, $A=\prod_i A_i$, and $D=\prod_i D_i$, with empty products 1.

Let $l,B,E\in\mathbb N$ and $n\ge3$. Assume $E T^l(T^K(n))<B(T^K(n)+3)$ and $2AB\le DE$. Assume also $\mathcal C(m)$ for every $0<m<n$. Then

The segment and terminal envelopes and cumulative budget are hypotheses; no all-root coverage is asserted.