Do all six-dimensional superconformal theories arise from F-theory
In plain words
A catalog of six-dimensional theories was built from geometric constructions in string theory. Whether every such theory appears in the catalog is unknown.
Precise statement
Heckman, Morrison, Rudelius and Vafa (2015) classified $6\mathrm{D}\,(1,0)$ SCFTs realized by F-theory on elliptically fibered Calabi-Yau threefolds with contractible bases. Determine whether every interacting $6\mathrm{D}\,(1,0)$ SCFT admits such a realization, including frozen phases with $\mathrm{O7}+$ planes, by a field-theoretic completeness argument or a counterexample.
What would settle it
A field-theoretic proof that consistency (anomaly cancellation and unitarity on the tensor branch) admits only the geometric list, or an explicit theory outside it.
Status in the literature
Frozen-singularity theories (Bhardwaj, Morrison, Tachikawa, Tomasiello 2018) enlarged the F-theory list; no field-theoretic completeness argument exists.