Nonfillability conjecture for contact structures from Seifert surgery diagrams

Let $M$ be a Seifert fibered $3$--manifold described by Figure~, and let the diagram define contact structures by contact $(\pm 1)$--surgeries according to the stated algorithm. Assume the coefficients $r_i$ satisfy the conditions~. A contact structure is symplectically fillable if it is the boundary contact structure of a compact symplectic four--manifold with the boundary orientation and compatibility condition described in the source.

Nonfillability conjecture. No contact structure defined by Figure~ and satisfying conditions~ is fillable.

The paper gives evidence for this claim and proves nonfillability for a substantial special family, namely certain tight structures with $k=1$ and $r_1=(\alpha+1)/(2\alpha+1)$. The general assertion remains open.