Let us consider $GL(V)$ over complex numbers. Let $P \subset G$ be a parabolic subgroup. Let $T \subset P$ be a maximal subtorus. Torus $T$ gives us a decomposition of $V$ into the sum of one-dimensional vector spaces. Let $e_{i}$ denote some basis of $V$ such that $e_{i}$ are eigenvectors of $T$. Let us fix some order $\geq$ on the set $\{e_{i}\}$. This order naturally corresponds to the regular cocharacter of $T$ (acts on the first element via $t$, on the second via $t^{2}$ and so on). Let us denote this cocharacter by $\nu$. It induces the action of $\mathbb{C}^{*}$ on the parabolic flag variety $G/P$. Thus we get a parabolic Bruhat order $\geq_{Br}$ on the set $(G/P)^{T}$ (take $p_{1}, p_{2} \in (G/P)^{T}$ then we say that $p_{1} \geq_{Br} p_{2}$ if $A_{p_{2}} \subset \overline{A}_{p_{1}}$, where $A_{p}$ denotes the attractor to a point $p$ via the action of $\nu$).
My question is the following: suppose now that we have two partial flags of type $P$ that are fixed under the action of $T$. Let us denote them $F,G$. Suppose also that for each $i$, $F_{i}$ is spanned by $e_{j_{1}}, \dots , e_{j_{i}}$ while $G_{i}$ is spanned by $e_{k_{1}}, \dots , e_{k_{i}}$ such that $e_{j_{l}} \geq e_{k_{l}}$ for every $l$ (here we mean that for any $i$ we can reorder basis elements of $F_{i}, G_{i}$ in such way). Is that true that $F \geq_{Br} G$?