Perron-number conjecture for growth rates of Coxeter systems of dimension at most two

A Coxeter system is a pair $(\Gamma,S)$, and its growth rate is the exponential growth rate of the number of elements of $\Gamma$ by word length with respect to $S$. The dimension is the dimension of the associated Davis complex. A Perron number is an algebraic integer $\tau>1$ that is either an integer or whose other Galois conjugates all have absolute value strictly less than $\tau$.

Perron-number conjecture. The growth rate of any Coxeter system of dimension at most $2$ is a Perron number.

The claim is motivated by the paper's results showing that growth rates in dimension at most two with positive or vanishing Euler characteristic are Salem or Pisot numbers, respectively, and that infinitely many systems with negative Euler characteristic have Perron growth rates. Whether every Coxeter system of dimension at most two has Perron growth rate remains open.