A series of word-hyperbolic Coxeter groups

For each positive integer $k$ we present an example of Coxeter system $(G_k,S_k)$ such that $G_k$ is a word-hyperbolic Coxeter group, for any two generating reflections $s,t\in S_k$ the product $st$ has finite order, and the Coxeter graph of $S_k$ has negative inertia index equal to $k$. In particular, $G_k$ cannot be embedded into pseudo-orthogonal group $O(n,k-1)$ for any $n$ as a reflection group with generating set being reflections w.r.t. $(G_k,S_k)$