Glover–Homer conjecture for real and complex generalized flag manifolds

Let $n_1,\ldots,n_k$ be distinct positive integers, let $n=n_1+\cdots+n_k$, and for $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$ define

$$\mathbb{F}M(n_1,\ldots,n_k)=\frac{U_{\mathbb{F}}(n)}{U_{\mathbb{F}}(n_1)\times\cdots\times U_{\mathbb{F}}(n_k)},$$

where $U_{\mathbb{R}}(n)=O(n)$ and $U_{\mathbb{C}}(n)=U(n)$. The fixed point property (f.p.p.) means that every selfmap has a fixed point. Glover–Homer's real–complex conjecture. If at most one of $n_1,\ldots,n_k$ is odd, then $\mathbb{F}M(n_1,\ldots,n_k)$ has the f.p.p., for $\mathbb{F}=\mathbb{R}$ and $\mathbb{F}=\mathbb{C}$. The conjecture proposes sufficiency for the parity condition accompanying the known necessary distinctness condition. The supplied source states that projective spaces and a family of complex flag manifolds were already known cases, but gives no proof or resolution of the full assertion.