The character identity for roots and the Orlik–Solomon representation

Let $W$ be a finite Coxeter group of rank $n$, with $R$ its representation on the set of roots. Let $G'$ be the virtual $W$-module obtained by evaluating at $t=1$ the graded quotient of the cohomology representation of the complexified hyperplane arrangement by $1-t$, and let $f_W$ be the integer appearing in the preceding dimension formula. Write $\operatorname{Reg}$ for the regular representation of $W$. Character identity. One should have an equality of characters

$$R\otimes G'=(-1)^{n-1}f_W\operatorname{Reg}.$$

The identity would refine the corresponding equality of dimensions and seek a representation-theoretic interpretation of the counting formula for full reflections. Its status is not resolved in the supplied text.