7-dimensional simply-connected spin manifolds whose integral cohomology rings are isomorphic to that of ${\mathbb{C}P}^2 \times S^3$ admit round fold maps

We have been interested in understanding the class of 7-dimensional closed and simply-connected manifolds in geometric and constructive ways. We have constructed explicit fold maps, which are higher dimensional versions of Morse functions, on some of the manifolds, previously. The studies have been motivated by studies of {\it special generic} maps, higher dimensional versions of Morse functions on homotopy spheres with exactly two singular points, characterizing them topologically except $4$-dimensional cases. The class contains canonical projections of unit spheres for example. This class has been found to be interesting, restricting the topologies and the differentiable structures of the manifolds strictly: Saeki, Sakuma and Wrazidlo found explicit phenomena. The present paper concerns fold maps on $7$-dimensional closed and simply-connected spin manifolds whose integral cohomology rings are isomorphic to that of the product of the $2$-dimensional complex projective space and the $3$-dimensional sphere.