The $q$-analog $3x+1$ orbit conjecture for $T_q$

Let $\mathbb{Z}_2$ denote the $2$-adic integers, let $F_2[[q]]$ be the ring of formal power series over $F_2$, and let $T$ and $T_q$ be the maps in the source. A conjugacy is a bijection $h\colon\mathbb{Z}_2\to F_2[[q]]$ satisfying $h\circ T=T_q\circ h$. The $q$-analog $3x+1$ orbit conjecture. For every such conjugacy $h$ and every positive integer $n$, the $T_q$-orbit of $h(n)$ contains $-3/7$. The conjugacy formulation transfers the classical $3x+1$ problem to the $q$-series dynamical system; whether the asserted orbit property holds is left open in the source.