The polynomial-correspondence conjecture for $T_{1,1+q^2}$

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 $T_{1,1+q^2}$ be the maps in the source. A conjugacy is a bijection $H\colon\mathbb{Z}_2\to F_2[[q]]$ satisfying $H\circ T=T_{1,1+q^2}\circ H$. Polynomial-correspondence conjecture. For every such conjugacy $H$ and every positive integer $n$, the power series $H(n)$ is a polynomial. If true, this would suffice to prove the classical $3x+1$ conjecture by the polynomial-orbit result established earlier in the paper.