The conjecture on Camacho–Sad indices of homogeneous convex foliations

Let $d\geq 2$. For a homogeneous convex foliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$, let $\mathcal{CS}_d$ denote the set of Camacho–Sad indices at non-radial singularities on the line at infinity. 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, and let $\mathcal{M}_d$ be the corresponding set for critically fixed rational maps of degree $d$. The conjecture. For $2\leq d\neq 4,5,7$,

$$\mathcal{CS}_d=\\{(1-d)^{\pm 1}\\},$$

or, equivalently,

\mathcal{M}_d=\left\\{d,\frac{d}{d-1}\right\\}.

The conjecture predicts that these are the only possible Camacho–Sad indices, equivalently fixed-point multipliers, in the stated degrees; the exceptional degrees $4$, $5$, and $7$ are excluded by the formulation.