The even-and-odd sequence length conjecture for $x^n+x+1$

Let $A=x^n+x+1$ with $n\geq 4$, and let $s$ be the greatest positive integer such that

$$n-2^{s+1}\geq 0.$$

Consider the even and odd sequences associated with $A$ under the binary polynomial Collatz transformations. Sequence length conjecture. Both the even and odd sequences of $A$ have length $2^s+1$. The claim is presented alongside computational data for this special family and is not established in the source.