Let $Gr_{i,n}$ denote the Grassmannian of complex linear $i$-dimensional subspaces in the Hermitian space $\mathbb{C}^n$. Let $1\leq i<n/2$. Consider the Radon transform between space of functions on two different Grassmannians: $$R\colon C^\infty(Gr_{i,n})\to C^\infty(Gr_{n-i,n})$$ given by $(Rf)(E)=\int_{F\subset E} f(F) dF$ where the integration is with respect to the Haar measure on the Grassmannian of complex $i$-subspaces of $E$.
Is it true that $R$ is an isomorphism? A reference would be most helpful.
Remark. For real Grassmannians the corresponding question has positive answer, see Gelʹfand, I. M.; Graev, M. I.; Roşu, R. The problem of integral geometry and intertwining operators for a pair of real Grassmannian manifolds. J. Operator Theory 12 (1984), no. 2, 359–383.
