Rational homotopy of maps between certain complex Grassmann manifolds

Let $G_{n,k}$ denote the complex Grassmann manifold of $k$-dimensional vector subspaces of ${\mathbb{C}}^n$. Assume $l,k\le \lfloor n\mathrm{/}2\rfloor$. We show that, for sufficiently large $n$, any continuous map $h:G_{n,l}\to G_{n,k}$ is rationally null homotopic if $(i)1\le k<l,$ $(ii)2<l<k<2(l-1)$, $(iii)1<l<k$, $l$ divides $n$ but $l$ does not divide $k$.