The Easy Coefficients Conjecture for rotation-symmetric quadratic Boolean functions

Let $I$ be an index set, let $Q_{I,n}$ be the associated rotation-symmetric quadratic Boolean function, and set

$$M(I)=\max_{i\in I}i.$$

Let $w_1,w_2,\ldots$ be the sequence obtained by extending the recursion for the weights of $Q_{I,n}$ backwards from $n=2M(I)+1$ to $n=1$, with $w_n=\operatorname{wt}(Q_{I,n})$ for $n\geq 2M(I)+1$. Let $\delta_1,\ldots,\delta_{2^{M(I)}}$ be the list of irrational roots of the characteristic polynomial of the rules matrix for $Q_{I,n}$, with the distinct roots $\delta_1,\ldots,\delta_{m(Q(I))}$ of the minimal polynomial listed first, followed by duplicates of these roots. Easy Coefficients Conjecture. The resulting sequence satisfies

$$w_n=2^{n-1}-\frac{1}{2}\left(\delta_1^n+\cdots+\delta_{2^{M(I)}}^n\right),\qquad n=1,2,\ldots.$$

The conjecture gives a uniform closed form for the weight recursion of rotation-symmetric quadratic Boolean functions. It is proved for MRS quadratic functions, while the general case for the functions $Q_{I,n}$ described here is not resolved by the supplied text.