NOTE: The question has now been posted on MathOverflow: Dominance of $w\mu$ for dominant cocharacter
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ of rank $n$ and a Borel $B \supset T$ defining a set of simple roots $\Delta$. Let $W$ be the Weyl group and denote by $X_*(T)_\mathbb{Q}$ the cocharacter group. For $I \subset \Delta$ $$W^I=\{w \in W \mid l(s_\alpha w)>l(w) \text{ for all } \alpha \in I \}.$$
Now let $w \in W^I$ and $\mu \in X_*(T)$ being greater equal zero with respect to the dominance order, i.e. $\mu= \sum_{\alpha \in \Delta} n_{\alpha} \alpha^{\vee} $ with $n_\alpha \geq 0$ for all $\alpha$. By assumption on $\mu$ we have $$w\mu= \sum_{\alpha \in \Delta} m_{\alpha} \alpha^{\vee}$$ with $m_\alpha$ not necessarily all non-negative. Additionally $\mu$ lies in the positive Weyl chamber.
In Type A I make in low dimensions the observation that if $m_\alpha \geq 0$ for all $\alpha \in \Delta \backslash I$ then $w\mu \geq 0$.
Is it something already known or even false in general?
P.S. In case that is a question for Mathoverflow let me know.