Gronwall's conjecture on uniqueness of linearization for non-hexagonal 3-webs

Let $\mathcal W$ and $\mathcal W'$ be germs of linear $3$-webs on $(\mathbb C^2,0)$ with non-vanishing curvature, and let $\varphi:(\mathbb C^2,0)\to(\mathbb C^2,0)$ be a germ of biholomorphism sending $\mathcal W$ to $\mathcal W'$. A germ of biholomorphism is projective when it is the germification of a projective automorphism of $\mathbb P^2$. Gronwall's conjecture. The germ $\varphi$ is the germification of a projective automorphism of $\mathbb P^2$. Equivalently, a non-hexagonal $3$-web admits at most one linearization. This conjecture concerns the uniqueness of linearization for linear webs; the supplied source states that it remains unsettled.