Bean's conjecture on the maximum of the normalized area of binary forms

Let $F$ be a binary form of degree $n\geq 3$ with complex coefficients and non-zero discriminant $D_F$, and define

$$Q(F)=|D_F|^{1/(n(n-1))}A_F.$$

Let

$$F_n^*(x,y)=\prod_{k=1}^n\left(\sin\left(\frac{k\pi}{n}\right)x-\cos\left(\frac{k\pi}{n}\right)y\right).$$

Bean's conjecture. The maximum value $M_n$ of $Q(F)$ over all such forms is attained precisely when $F$, up to multiplication by a complex number, is equivalent under $\operatorname{GL}_2(\mathbb R)$ to $F_n^*$. Equivalently,

$$M_n=Q(F_n^*).$$

Moreover,

$$\lim_{n\to\infty}M_n=2\pi.$$

The quantity $Q(F)$ is invariant under multiplication by a complex number and under the action of $\operatorname{GL}_2(\mathbb R)$, making this a natural extremal problem for binary forms. The source attributes this conjecture to Bean; no resolution is given here.