The $\frac{5}{4}$-conjecture for even four-manifolds

Let $X$ be a closed oriented even $4$-manifold, meaning that its intersection form is even, and let $\sigma(X)$ denote its signature. The $\frac{5}{4}$-conjecture. One has

$$\frac{5}{4}|\sigma(X)|\leq b_2(X).$$

This is a weak analogue of the $\frac{11}{8}$-conjecture for spin $4$-manifolds. It extends inequalities known in special cases, including Furuta's inequality for closed oriented spin $4$-manifolds with $b_2(X)>0$ and Bohr's result for certain fundamental groups; the general statement remains open.