Polynomial area growth for special Kähler metrics

Let $g=w|dz|^2$ be an affine special Kähler metric on a punctured disc and let $Ξ=q(z)\,dz^3$ be its associated holomorphic cubic differential. We prove that if the $g$-area of concentric punctured neighbourhoods grows at most polynomially, then $q$ is meromorphic at the puncture. More precisely, an