Kummer-type constructions of almost Ricci-flat 5-manifolds

A smooth closed manifold $M$ is called almost Ricci-flat if $$\inf_g||\textrm{Ric}_g||_\infty\cdot \textrm{diam}_g(M)^2=0$$ where $\textrm{Ric}_g$ and $\textrm{diam}_g$ denote the Ricci tensor and the diameter of $g$ respectively and $g$ runs over all Riemannian metrics on $M$. By using Kummer-type method we construct a smooth closed almost Ricci-flat nonspin 5-manifold $M$ which is simply connected. It's minimal volume vanishes, namely it collapses with sectional curvature bounded.