A Fibonacci theorem for Collatz trajectories via modular graph structure

Let $T(n)=n/2$ if $n$ is even and $T(n)=(3n+1)/2$ if $n$ is odd. We prove that for each $m\ge1$, exactly $F(m+1)$ odd integers $n$ in $\{1,\ldots,2^m\}$ have the property that none of the iterates $T(n),T^2(n),\ldots,T^{m-1}(n)$ lies in the residue class $4\pmod6$, where $F(m+1)$ is the $(m+1)$-th F