In some other context, the question was raised how many nearly Kähler structures exist on the sphere ${𝕊}^6$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a consequence of the description of the eigenspace to the eigenvalue $\lambda =12$ of the Laplacian acting on $2$-forms. A similar result concerning nearly parallel ${\mathrm{G}}_2$-structures on the round sphere ${𝕊}^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.