We show that some embedded standard $13$-spheres in Shimada's exotic $15$-spheres have $\mathbb{Z}_2$ quotient spaces, $P^{13}$s, are fake real $13$-dimensional projective i.e., they homotopy equivalent, but not diffeomorphic to the $\mathbb{R}\mathrm{P}^{13}$. As observed by F. Wilhelm and second named author [RW], Davis $\mathsf{SO}(2)\times \mathsf{G}_2$ actions on descend cohomogeneity one ...