Infinity-squared property for the positive-signature orientation of K3

Determine whether the orientation-reversed K3 surface \overline{E(2)}, whose intersection form is 2E_8\oplus 3H and whose signature is positive, has the \infty^2-property and therefore supports a symplectic structure in the relevant sense.

Background

The K3 surface E(2) has the \infty2-property, but reversing its orientation produces the positive-signature spin manifold \overline{E(2)}. The paper observes that the existence of a symplectic structure on this orientation reversal is not known.

Consequently, it is also unknown whether \overline{E(2)} has the stronger \infty2-property. This example motivates the broader study of positive-signature spin symplectic geography.

References

Then similar to above, $\overline{E(2)}$ (with $\sigma(\overline{E(2)})=-\sigma(E(2))=16$) will also have $\infty$-property, but it is unknown at the moment whether $\overline{E(2)}$ supports any symplectic structure. Hence we do not know at the moment whether the positive-signature spin $4$-manifold $\overline{E(2)}$ has $\infty2$-property.

— Slope Inequalities for the Geography Problem of Spin Symplectic 4-Manifolds  (2608.25889 - Fushida-Hardy et al., 26 Aug 2026) in Section 1, paragraphs following Definition 1.1