Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra, $w \in W$ an element of the Weyl group, and $\lambda$ an integral dominant weight with $V(\lambda)$ the associated irreducible highest weight representation. Then we can consider the Demazure module $$ V_w(\lambda):= U(\mathfrak{b}).v_{w\lambda} $$ where $\mathfrak{b} \subset \mathfrak{g}$ is the standard Borel subalgebra and $v_{w\lambda} \in V(\lambda)$ is a nonzero extremal vector of weight $w\lambda$.
The character of $V_w(\lambda)$, which captures the dimensions of the weight spaces $V_w(\lambda)_\mu$, is famously given by the Demazure character formula, or can be expressed in the combinatorial language of Littelmann paths and Demazure crystals. It is also known, for a fixed weight $\mu$, that there exists a $w \in W$ sufficiently "high up" the Bruhat order (but still of finite length) such that $$ \text{dim} (V_w(\lambda)_\mu) = \text{dim} (V(\lambda)_\mu). $$
Of course, if $\mathfrak{g}$ is finite-dimensional semisimple, we can take the longest element $w_0$ so that $V_{w_0}(\lambda)=V(\lambda)$ and we get the above equality for all $\mu$, but this is not applicable in the general symmetrizable case.
Question: Are there results which give a way to determine, for a fixed $\lambda$ and $\mu$, a Weyl group element $w$ such that $\text{dim} (V_w(\lambda)_\mu) = \text{dim} (V(\lambda)_\mu)$?
I am particularly interested in the following scenario: fix $\beta$ a real root of $\mathfrak{g}$, and let $s_\beta \in W$ be the associated reflection. I would like to compare the weight spaces $V_{s_\beta}(\lambda)_{\lambda-\beta}$ and $V(\lambda)_{\lambda-\beta}$; we can take the simplifying assumption that $\lambda$ is regular dominant for now, if that is helpful.
