The binary polynomial Collatz conjecture

Let $A\in\mathbb{F}_2[x]\setminus\{0\}$. Define the associated sequences $(a_{2k})_k$, $(A_{2k})_k$, and $(A_{2k+1})_k$ by $A_0=A$, $A_{2k}=1+(x^2+x+1)A_{2k-1}$, and by removing all powers of $x$ and $x+1$ from each even polynomial to obtain the following odd polynomial. Binary polynomial Collatz conjecture. For a given $A\in\mathbb{F}_2[x]\setminus\{0\}$, there exists $m\in\mathbb{N}^*$ such that for all $k\geq m$, $A_{2k}=x(x+1)$ and $A_{2k+1}=1$. This is the proposed analogue over binary polynomials of the integer Collatz conjecture; the source does not provide a resolution.