Holomorphy of the curvature of planar webs along an invariant curve and applications to the flatness of certain webs on $\mathbb{P}^{2}_{\mathbb{C}}$

Let $\mathcal{W}=\mathcal{W}_{n}\boxtimes\mathcal{W}_{d-n}$ be a $d$-web on $(\mathbb{C}^2,0)$, where $\mathcal{W}_{n}$ is an $n$-web admitting an irreducible totally invariant curve $C$, and $\mathcal{W}_{d-n}$ is a regular $(d-n)$-web transverse to $C$. We show that the curvature of $\mathcal{W}$