The repeated-degree conjecture for the family $x^n+x+1$

Let $A=x^n+x+1$ with $n\geq 2$, and let $A_1,A_3,\ldots$ be the odd subsequence in the binary-polynomial Collatz transformation. Let $s$ be the greatest integer such that $n-2^{s+1}\geq 1$. Repeated-degree conjecture. For any positive integer $t\leq s-1$, the odd sequence contains $2^t$ polynomials having the same degree $d_t$. In particular,

$$d_1=\deg(A_5)=\deg(A_7)$$

and

$$d_2=\deg(A_9)=\deg(A_{11})=\deg(A_{13})=\deg(A_{15}).$$

This is one of two conjectures stated for the family $x^n+x+1$. It describes a pattern in the degrees of odd iterates; the supplied text gives examples but no resolution.