Garner's conjecture on corresponding Collatz stems

Let $T$ be the accelerated Collatz map, and let $s_i$ and $s_i'$ be the parity vectors

$$s_i=\langle 0,\underbrace{1,1,\ldots,1}_{i\text{ ones}},0,1\rangle,$$ $$s_i'=\langle 1,\underbrace{1,1,\ldots,1}_{i\text{ ones}},0,0\rangle.$$

A pair of parity sequences $s$ and $s'$ are corresponding stems if, for every integer $x$, $T_s(x)=T_{s'}(x+1)$, while equal-length initial subsequences $v,v'$ satisfy $|T_v(x)-T_{v'}(x+1)|\ne 1$ and $T_v(x)\ne T_{v'}(x+1)$.

Garner's conjecture. Every pair of corresponding stems is of the form $s_i$ and $s_i'$ for some $i$; equivalently, any pair of consecutive integers of the same height has parity vectors for the non-overlapping parts of their trajectories ending in $s_i$ and $s_i'$.

The conjecture is intended to characterize consecutive integers with equal Collatz height through their parity vectors. The paper states that it has counterexamples, so this claim is refuted.