The degree multiplicity 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

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

For each positive integer $t\leq s-1$, denote by $d_t$ the common degree asserted for the relevant terms of the odd sequence. Degree multiplicity conjecture. 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 a computationally motivated structural claim for the family $x^n+x+1$; the source gives no proof or resolution.