The binary-form area lower-bound conjecture

Let $F$ be a binary form with complex coefficients of degree $n\geq 1$, and let $r$ be the number of its roots on the real projective line $\mathbb R\mathbb P^1$, counted with multiplicity. Define

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

Binary-form area lower-bound conjecture. One has

$$h_F^{2/n}A_F\geq h_{F_{n,r}}^{2/n}A_{F_{n,r}}.$$

This conjecture proposes that, among degree-$n$ binary forms with exactly $r$ real projective roots, the displayed family minimizes the scale-invariant quantity $h_F^{2/n}A_F$. The preceding examples show the claimed extremal behavior for the cases $r=0$ and asymptotically for $r=n$, but the general assertion is presented as a conjecture.