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

Let's say $G$ is a semisimple algebraic group over an algebraic field of characteristic 0 and $\lambda$ is a dominant integral weight. Consider the irreducible finite dimensional 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" (see below) in the Weyl group of $G$.

Here $m_{\lambda\mu}$ is the dimension of $\mu$ weight space in $V_\lambda$. The meaning of "Coxeter elements" is as follows: fix a set of simple reflections $s_1,...,s_r$ in the Weyl group $W$, by a "Coxeter element" I mean an element of $W$ which can be written as products of *these* $r$ simple reflections, each occurring precisely once, in any order. As Jim Humphreys points out, this is not the usual definition of Coxeter elements since it depends on the choice of simple reflections, but the number of such elements does not depend on this choice.

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.