The lower-bound conjecture for Camacho–Sad indices of homogeneous convex foliations

Let $\mathcal{H}$ be a homogeneous convex foliation of degree $d\geq2$ on $\mathbb{P}^{2}_{\mathbb{C}}$, and let $s\in\ell_\infty$ be a non-radial singularity of $\mathcal{H}$. For a rational map $f\colon\mathbb{P}^{1}_{\mathbb{C}}\to\mathbb{P}^{1}_{\mathbb{C}}$, let $\mathcal{M}(f)$ be the set of its nonzero, nonunit fixed-point multipliers. The lower-bound conjecture. One has

$$\frac{1}{d-1}\leq-\operatorname{Re}\bigl(\operatorname{CS}(\mathcal H,\ell_\infty,s)\bigr).$$

Equivalently, if $f$ is a critically fixed rational map of degree $d\geq2$, then $\mathcal{M}(f)$ is contained in the closed disk

$$\overline{\mathbb{D}}\left(\frac{d+1}{2},\frac{d-1}{2}\right)\subset\mathbb{C},$$

of center $\frac{d+1}{2}$ and radius $\frac{d-1}{2}$. This would improve the currently stated lower bound for the real part of the Camacho–Sad index, and the source notes that the proposed value is attained in known examples.