The 10/8-conjecture for non-spin 4-manifolds

Let $X$ be a smooth closed oriented non-spin 4-manifold with even intersection form

$$kE_8\oplus nH.$$

The $10/8$-conjecture. Then

$$n\geq |k|.$$

This conjecture concerns restrictions on the intersection forms of smooth non-spin 4-manifolds. It is motivated by Furuta's $10/8$-theorem for spin 4-manifolds and by partial results obtained using covering tricks under conditions on the torsion part of $H_1(X,\mathbb{Z})$.