  We generalise recent results of Afsar, Larsen and Neshveyev for product
systems over quasi-lattice orders by showing that the equilibrium states of
quasi-free dynamics on the Nica-Toeplitz $C^*$-algebras of product systems over
right LCM monoids must satisfy a positivity condition encoded in a system of
inequalities satisfied by their restrictions to the coefficient algebra. We
prove that the reduction of this positivity condition to a finite subset of
inequalities is valid for a wider class of monoids that properly includes
finite-type Artin monoids, answering a question left open in their work. Our
main technical tool is a combinatorially generated tree modelled on a recent
construction developed by Boyu Li for dilations of contractive representations.
We also obtain a reduction of the positivity condition to inequalities arising
from a certain minimal subset that may not be finite but has the advantage of
holding for all Noetherian right LCM monoids, and we present an example,
arising from a finite-type Artin monoid, that exhibits a gap in its inverse
temperature space.
