Shokurov global index conjecture for threefold foliations

The conjecture asks whether numerically trivial log-canonical foliated log Calabi–Yau triples in dimension three have a uniformly bounded global index. The paper describes this as a question of Meng, Xie, and one of its authors.

Known results

• $\mathcal{F}=T_X$: the classical conjecture in dimension at most $3$ (Xu, 2020, based on Jia, 2021).
• Nonzero boundary $B$: proved by LLM, 2023.
• Dimension at most $2$: proved by LLM, 2023, based on Pereira, 2005.
• The non-effective version was known in dimension at most $3$, for rank-one foliations, and for algebraically integrable foliations.

May 2026 affirmative preprint

The paper claims that for any DCC coefficient set $\mathcal{I}\subset[0,1]capmathbb{Q}$, there is an $I$ depending only on $\mathcal{I}$ with $I(K_{\mathcal{F}}+B)\sim0$ in dimension at most $3$. It also claims $Ile30$ for canonical, non-algebraically-integrable foliations. The claim is unverified; the authors say the main result was partially obtained using Rethlas.

Current status (as of May 2026): The conjecture is claimed proved for dimension at most $3$ by an arXiv preprint, while independent verification remains outstanding.