This is a technical lemma I am currently stuck at. Any suggestions about how to proceed are welcome.

Let $G$ be a split semisimple group over a number field. Let $B, T$ be a chosen minimal parabolic, torus respectively, and $\theta$ be an automorphism of $G$ of finite order, preserving $B$ and $T$. By the theory of algebraic groups, we can associate to $(G, B, T)$, a root datum $(X^*, \Delta, X_*, \Delta^\vee )$, where $X^*$ (resp. $X_*$) is the lattice of characters (cocharacters) and $\Delta$ (resp. $\Delta^\vee$) the simple roots (coroots). The Weyl group $W$ acts on the root space $V^* = X^* \otimes \mathbb R$ and co-root space $V_*$ as usual and the action of $\theta$ descends to a permutation of the finite sets $\Delta, \Delta^\vee$. For every $\beta \in \Delta$, let $\varpi_\beta \in \hat\Delta$ be the corresponding weight so that $\hat\Delta$ is a basis of $V^*$ dual to $\Delta$.

Question: Fix $w \in W, w \neq 1$. Does there exist a cone $\Omega$ inside the positive Weyl chamber of $V^*$ such that if $\lambda \in \Omega$ and $\lambda - \theta w \lambda = \displaystyle\sum_{\beta \in \Delta} {d_\beta} \beta$, then $d_\beta > 0$?

(Edit: To clarify, by positive Weyl chamber, I mean a positive span of weights, not roots.)

This is not true per se but I would like the coordinates $d_\beta$ positive only in some directions: Let $Q(w)$ be the smallest standard (i.e., containing $B$) parabolic subgroup containing a representative of $w$. By a correspondence between standard parabolic subgroups and subsets of $\Delta$, we have the subset $\Delta^{Q(w)}$ of $\Delta = \Delta^G$ corresponding to $Q(w)$. I need $d_\beta>0$ whenever $\beta \in \Delta^{Q(w)}$.

Example: Let $G = SL(n)$ with usual Borel and torus and let $\theta(x) = {}^Tx^{-1}$. We can identify $V^*$ with the subspace of $\mathbb R^n$ of vectors with coordinates adding to zero so that $\Delta = \{ \beta_1 = (1, -1, 0, \cdots, 0), \cdots, \beta_{n-1} \}$ and $\{ \varpi_\beta \in \hat\Delta\}$ is the dual basis (of weights). We have $\theta(\beta_i) = \beta_{n-i}$. In the case of $SL(3)$ taking $\lambda = c_1 \varpi_1+ c_2 \varpi_2$ and $w$ to be the permutation $(1,2)$ gives $$ \lambda - \theta w\lambda = \left(\frac{c_1 - c_2}{3}\right)\beta_1 + \left(\frac{2c_1 + c_2}{3}\right)\beta_2 $$ so we can define the cone to be $c_1 > c_2 > 0$. I have written a Python code and verified this till $SL(7)$ but unable to prove so far for general $G$.

EDIT: When $\theta = 1$ we can take $\Omega$ to be the full positive Weyl chamber; see Bourbaki's Lie Groups and Lie Algebras Chapter 6 $\S 1.5$ (Edit: 1.6, not 1.5) Proposition 18 for a proof based on induction on $\ell(w)$.