Three new almost positively curved manifolds

A Riemannian manifold is called almost positively curved if the set of points for which all $2$-planes have positive sectional curvature is open and dense. We find three new examples of almost positively curved manifolds: $Sp(3)\mathrm{/}Sp(1{)}^2$, and two circle quotients of $Sp(3)\mathrm{/}Sp(1{)}^2$. We also show the quasi-positively curved metric of Tapp [26]} on $Sp(n+1)\mathrm{/}Sp(n-1)Sp(1)$ is not almost positively curved if $n\ge 3$.