Conjecture on regular maps between homogeneous projective varieties

Let $X=G/P$ and $X'=G'/P'$ be homogeneous projective varieties, where $G$ and $G'$ are connected simple algebraic groups and $P$ and $P'$ are parabolic subgroups. For a realization of a homogeneous projective variety as $G/P$, define its minimum semisimple stabilizer rank (respectively, maximum semisimple stabilizer rank) to be the minimum (respectively, maximum) of the ranks of the semisimple parts of the Levi components of $P$ over all such realizations; denote these quantities by minss rank $X$ and maxss rank $X$. Let $\bbP^{2n}$ denote projective space, with $n\geq 1$. Conjecture. (a) If $X$ is different from $\mathbb{P}^{2n}$ and

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

then there does not exist any non-constant regular map from $X$ to $X'$. (b) If $X=\mathbb{P}^{2n}$ and there exists a non-constant regular map from $X$ to $X'$, then

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

The conjecture proposes numerical criteria governing the non-existence of non-constant regular maps between homogeneous projective varieties; the source provides no resolution status or supporting evidence beyond posing it as a conjecture.