Kumar's conjecture on morphisms between homogeneous projective varieties

Let $X$ and $X'$ be connected indecomposable homogeneous projective varieties. For such a variety, write $\operatorname{minss\ rank} X$ and $\operatorname{maxss\ rank} X$ for its minimum and maximum semisimple stabilizer ranks over all realizations as $H/P$, where $H$ is a simple connected algebraic group and $P$ is parabolic. Kumar's conjecture. (a) If $X\neq\mathbb{P}^{2n}$ for $n\geq 1$ and

$$\operatorname{minss\ rank} X>\operatorname{maxss\ rank} X',$$

then there is no nonconstant algebraic map from $X$ to $X'$. (b) If $X=\mathbb{P}^{2n}$ for $n\geq 1$ and there exists a nonconstant regular map from $X$ to $X'$, then

$$\operatorname{minss\ rank}\mathbb{P}^{2n-1}=n-1\leq\operatorname{maxss\ rank} X'.$$

The conjecture proposes numerical obstructions to nonconstant morphisms between indecomposable homogeneous projective varieties, with a separate exceptional condition for even-dimensional projective spaces.