Modified Levenberg–Martin–Poletsky conjecture on pluripolar hulls of finely holomorphic graphs

Let $U$ be a fine domain in $\mathbb{C}$, and let $f$ be finely holomorphic on $U$. Denote by $\Gamma_f^*$ the pluripolar hull of the graph of $f$, and by the maximal finely holomorphic continuation of $f$ its graph over the corresponding maximal fine domain. Modified Levenberg–Martin–Poletsky conjecture. The pluripolar hull $\Gamma_f^*$ equals the graph of the maximal finely holomorphic continuation of $f$. The conjecture asserts that the examples described above exhaust the possible points of the pluripolar hull: every point in the hull should arise from the maximal finely holomorphic continuation.