Le tissu dual d'un pré-feuilletage convexe réduit sur ${\mathbb{P}}_{\mathbb{C}}^2$ est plat

A holomorphic pre-foliation $\mathcal{F}=\mathcal{C}⊠\mathcal{F}$ on ${\mathbb{P}}_{\mathbb{C}}^2$ is the data of a reduced complex projective curve $\mathcal{C}$ of ${\mathbb{P}}_{\mathbb{C}}^2$ and a holomorphic foliation $\mathcal{F}$ on ${\mathbb{P}}_{\mathbb{C}}^2$. When the foliation $\mathcal{F}$ is convex (resp. reduced convex) and the curve $\mathcal{C}$ is invariant by $\mathcal{F}$, we say that the pre-foliation $\mathcal{F}=\mathcal{C}⊠\mathcal{F}$ is convex (resp. reduced convex). We prove that the dual web of a reduced convex pre-foliation on ${\mathbb{P}}_{\mathbb{C}}^2$ is flat. This generalizes our previous result obtained in the case where the associated curve consists only of invariant lines.