The Collatz function $T$ is defined on the set $\mathbb{Z}^+$ of positive integers as: $T(n)=n/2$ if $n$ is even, and $T(n)=3n+1$ if $n$ is odd. Let $T^k$ be the $k$th iteration of $T$. We say $n$ terminates if $T^k(n)=1$ for some $k$.
Let $n$ be an integer of the form $$3^{2^k(j-1)}+3^{2^k(j-2)}+\cdots +3^{2^k}+1$$
where $k,j\in \mathbb{Z}^+$ and $j$ is odd.
Question: Will $n$ terminate for all such $k,j$?