The length conjecture for the family $x^n+x+1$

Let $A=x^n+x+1$ with $n\geq 2$, and let $\ell_A$ denote the length of its binary-polynomial Collatz transformation sequence. Let $s$ be the greatest integer such that $n-2^{s+1}\geq 1$. Length conjecture for $x^n+x+1$. The length $\ell_A$ equals

$$2^s+1.$$

This is the second conjecture stated for the family $x^n+x+1$, following the conjectured repeated-degree pattern. The supplied text does not state a proof or resolution.