This question concerns lower bounds of certain weight multiplicities in representations of algebraic groups (or Lie groups, Lie algebras).

Let's say $G$ is a semisimple algebraic group and $\lambda$ is a dominant weight. Consider the irreducible representation $V_\lambda$ with highest weight $\lambda$ and let $\mu$ be a weight occuring in $V_\lambda$.

Suppose that $\lambda$ is regular (lies in interior of Weyl chamber), $\mu$ is dominant, and $\lambda-\mu$ lies in the interior of the positive root cone (the "wide cone"), then it seems to be true that 

$m_{\lambda\mu}\ge$ the number of Coxeter elements in the Weyl group of $G$.

Here $m_{\lambda\mu}$ is the dimension of $\mu$ weight space in $V_\lambda$. Coxeter elements are defined as follows: fix a set of simple reflections $s_1,...,s_r$ in the Weyl group $W$, a Coxeter element is an element of $W$ which can be written as products of these $r$ simple reflections, each occurring precisely once, in any order.

My questions is: Does anyone know an elementary or straightforward proof (or maybe counterexample) of this lower bound? 
For example using some multiplicity formula or combinatorial models.

I came to this possible lower bound in a very indirect way when studying geometry of certain affine Springer fibers. But I guess there should be a straightforward way to see this, for example by some weight multiplicity formula, which I'm not quite familiar with. It would also be great if this could be seen by using some combinatorial models of algebraic representations (crystals, path models, MV polytopes etc). But since I'm not familiar with all these, any comments or suggestions are welcome.