Consider the following points:

$\bullet$ Let ${\cal Harm}(n,d)$ denote the harmonic forms of the de Rham complex of the Grassmannian $Gr_{\mathbb{C}}(n,d)$ with respect to the Riemannian metric induced by the Fubini--Study metric $g$ of complex projective space.

$\bullet$ By Hodge decomposition we have an isomorphism between ${\cal Harm}(n,d)$ and $H^{\bullet}(n,d)$, the cohomology ring of $Gr_{\mathbb{C}}(n,d)$.

$\bullet$ Let $(\cdot,\cdot)$ denote the inner product induced on ${\cal Harm}(n,d)$ by $g$ composed with integration with respect to the Haar measure, and by abuse of notation the inner product induced on the cohomology ring $H^{\bullet}(n,d)$.

What I wonder is whether or not the standard Schubert cell basis of $H^\bullet(n,d)$ (as explained for example in this M.O. answer) is orthogonal, or even orthonormal with respect to this inner product?