Matsumoto's 11/8-conjecture for spin 4-manifold intersection forms

Let $p$ be a nonnegative integer and $q$ a natural number. The form

$$2pE_{8}\oplus q\begin{pmatrix}0&1\\1&0\end{pmatrix}$$

can be realized as the intersection form of a closed smooth spin $4$-manifold. Matsumoto's 11/8-conjecture. The form can be realized if and only if $q\geq 3p$. This is the final missing condition in the geography problem for even indefinite intersection forms; the “if” direction is realized by connected sums of copies of $K3$ and $S^{2}\times S^{2}$, while the “only if” direction is the substantive open part, equivalent to the 11/8 inequality.