Meromorphic extension conjecture for pluripolar curves

Let $\gamma$ be a pluripolar curve, let $\hat{\gamma}$ denote its projective hull, and let $\Sigma$ be a set contained in $\hat{\gamma}$. Let $\phi$ be the function associated with $\Sigma$ in the preceding setup, with $\phi\in C(\mathbb{D}^*\cup\mathbb{T})$. Meromorphic extension conjecture. If

$$\Sigma\subset\hat{\gamma},$$

then $\phi$ is meromorphic in $\mathbb{D}$. Proposition establishes holomorphicity on $\mathbb{D}^*$ under the same pluripolarity assumption; the conjecture asks whether the function extends meromorphically across the puncture.