Exponential growth rates of even Coxeter groups

Let $W$ be an even Coxeter group.

We prove that among all Coxeter systems generating $W$ the unique even Coxeter system realizes the minimal exponential growth.

Our proof relies on comparing the exponential growth rates in the explicit algorithm of Mihalik which from any Coxeter system of an even Coxeter group eventually produces the unique even one.

The main new ingredient is that blow downs along pseudo-transpositions do not increase the exponential growth rate.