Conjectural classification of rational homology ball fillings of Seifert fibered spaces

Let $Y=Y(e_0;r_1,r_2,\ldots,r_n)$ be a Seifert fibered space that is also a rational homology sphere, and let $\xi$ be a contact structure on $Y$. Seifert filling classification conjecture. The pair $(Y,\xi)$ admits a symplectic rational homology ball filling if and only if $n\leq 4$ and one of the following holds: (1) $e_0\leq -2$, $Y$ is the link of a complex surface singularity whose resolution graph belongs to one of the finite families provided by Bhupal and Stipsicz, and $\pm\xi$ is the canonical contact structure; (2) $e_0=-1$, $Y$ is a small Seifert fibered space with complementary legs as in Theorem 1.6, and $\xi$ is one of the balanced contact structures described there; or (3) $e_0\geq 0$, $Y$ is a small Seifert fibered space with complementary legs as in the first part of Theorem 1.7, and $\xi$ is one of the four balanced contact structures described there. This conjecture proposes a general classification of Seifert fibered rational homology spheres admitting symplectic rational homology ball fillings, extending the cases established in the cited results and the paper's theorems.