Nonexistence of regular maps between homogeneous projective varieties

We study non-existence of non-constant regular maps from a partial flag variety $X=G\mathrm{/}P$ to another partial flag variety $X^{\mathrm{\prime }}=G^{\mathrm{\prime }}\mathrm{/}{P}^{\mathrm{\prime }}$ and prove that there does not exist any non-constant regular map from any partial non-complete flag variety $X$ to any complete flag variety $X^{\mathrm{\prime }}=G^{\mathrm{\prime }}\mathrm{/}{B}^{\mathrm{\prime }}$. We also formulate a general conjecture on the non-existence of non-constant regular maps from $X\to X^{\mathrm{\prime }}$.