Let $\left(W\text{, }S\right)$ be a Coxeter system. For every $w\in W$ let us write $\left|w\right|$ for the length of $w$. Set $\lambda\left(e\right)=1$ where $e\in W$ denotes the neutral element of the group and define coefficients $\lambda\left(w\right)\in\mathbb{Z}$ recursively via \begin{eqnarray} \sum_{v\in W \text{: }\left|v^{-1}w\right|=\left|w\right|-\left|v\right|}\lambda\left(v\right)=0 \end{eqnarray} for any $w\in W$. For example this gives us $\lambda\left(e\right)=1$, $\lambda\left(s\right)=\left(-1\right)$ for all $s\in S$, $\lambda\left(st\right)=1$ if $m_{st}=2$ and $\lambda\left(st\right)=0$ if $m_{st}\neq2$ (for the notation see Wikipedia) for all $s\text{, }t\in S$ with $s\neq t$, and so on.

My question is: Does there always exist some $l\in\mathbb{N}$ such that $\lambda\left(w\right)=0$ for all $w\in W$ with $\left|w\right|\geq l$?

In the case that $\left(W\text{, }S\right)$ is right-angled (i.e. $m_{st}\in\left\{ 2\text{, }\infty\right\}$ for all $s\text{, }t\in S$ with $s\neq t$) this is true and we can take $l=\left|S\right|$. I'm wondering if in the general case this property also holds. If no, is it possible to characterize the property in an instructive way?