Does there exist a sequence $A_n=A_n(x)\in\mathbb F_2[x]$ over the field $\mathbb F_2$ of two elements (represented by $0$ and $1$) such that $A_n(0)=A_n(1)=1$ and the inverse series $1/A_n=\sum_{j=0}^\infty \alpha_{n,j}x^j\in \mathbb F_2[[x]]$ have supports with densities $\delta_n=\lim_{k\rightarrow\infty} \frac{\alpha_{n,0}+\alpha_{n,1}+\dots+\alpha_{n,k-1}}{k}$ converging to $1$?

A positive answer to this question would give a positive answer to question [Sum of densities of support of $A$ and $A^{-1}$ for $A=1+\dots\in \mathbb F_2[[x]]$ by considering $\frac{A_n(x^2)}{1+x}$.

The highest possible density for polynomials of degree $\leq 16$ is $\frac{2}{3}$, achieved by $1+x+x^2$.