A non-uniform distribution property of most orbits, in case the $3x+1$ conjecture is true

Let $T(n)=\left{\begin{array}{ll}3n+1&(n\hbox{ odd})\frac n2&(n\hbox{ even})\end{array}\right.$($n\in\mathbb Z$)\mathrm{.}Wecall\mathrm{"}theorbitoftheinteger$n$\mathrm{"},theset$${\mathcal{O}}_n:=\{m\in \mathbb{Z}:\mathrm{\exists }k\ge 0,m=T^k(n)\}$$andweput$c_i(n):=#{m\in\mathcal O_n;:;m\equiv i\hbox{ mod.}18}$\mathrm{.}Let$W$bethesetoftheintegerswhoseorbitcontains$1$ and is, in the following sense, about well distributed modulo $18$ between the six elements of the set $I:=\{1,5,7,11,13,17\}$ (the elements of {1,\dots,18} that are odd and not divisible by $3$). More precisely: $$W:=\{n\in \mathbb{N}:\mathrm{\exists }k\ge 0,T^k(n)=1\text{ and }\mathrm{\forall }i\in I,\frac{c_i(n)}{\sum _{i\in I}{c}_i(n)}\le \frac{1}{6}+0.0215\}\mathrm{.}$$ We prove that $W$ has density $0$ in $\mathbb{N}$. Consequently, if the $3x+1$ conjecture is true, most of the positive integers $n$ satisfy $$\frac{\underset{i\in I}{\max }{c}_i(n)}{\sum _{i\in I}{c}_i(n)}>\frac{1}{6}+\mathrm{0.0215.}$$