Flower-support conjecture for maximizing measures of odd trigonometric polynomials

Let $n\in\mathbb{Z}_{>0}$ be odd. Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the circle, and let $E_2:\mathbb{T}\to\mathbb{T}$ be the doubling map. For observables

$$f(x)=a_1\cos(2\pi x)+b_1\sin(2\pi x)+a_3\cos(6\pi x)+b_3\sin(6\pi x)+\cdots+a_n\cos(2n\pi x)+b_n\sin(2n\pi x),$$

consider measures maximizing the integral of $f$ among all $E_2$-invariant probability measures. Flower-support conjecture. All maximizing measures are supported in a flower with at most $n$ petals. This conjecture extends the known degree-one trigonometric-polynomial result for the doubling map to higher odd degrees; the status of the proposed bound for general odd $n$ is open.