A stability-like theorem for cohomology of pure braid groups of the series A, B and D

Consider the ring $R:=\text{Q}[\tau ,{\tau }^{-1}]$ of Laurent polynomials in the variable $\tau$. The Artin’s Pure Braid Groups (or Generalized Pure Braid Groups) act over $R,$ where the action of every standard generator is the multiplication by $\tau$. In this paper we consider the cohomology of such groups with coefficients in the module $R$ (it is well known that such cohomology is strictly related to the untwisted integral cohomology of the Milnor fibration naturally associated to the reflection arrangement). We give a sort of $\mathit{\text{stability}}$ theorem for the cohomologies of the infinite series $A$, $B$ and $D,$ finding that these cohomologies stabilize, with respect to the natural inclusion, at some number of copies of the trivial $R$-module $\text{Q}$. We also give a formula which compute this number of copies.