In the article "Hodge theory for combinatorial geometries" by Adiprasito, Huh and Katz, it it claimed in the proof of theorem 5.12 that there is a Chow equivalence between the de Concini-Processi wonderful model $Y$ of an arrangement, and a certain toric variety $X$.

However, the source they cite only proves that

$$H^*(Y)\cong Ch(X)$$

It seems that to draw the conclusion that $Ch(Y)\cong Ch(X)$ it is used that $Ch(Y)\cong H^*(Y)$, but I do not know why that is the case?

Here is what I know: $Y\subset X$ as a closed subset, and the relevant maps are (should be) induced by this inclusion. We have a diagram $$ \require{AMScd} \begin{CD} Ch(X) @>{ch}>> H^*(X)\\ @VVV @VVV \\ Ch(Y) @>{ch}>> H^*(Y) \end{CD} $$ and we know that the diagonal arrow is an isomorphism. To establish that $Ch(X)\cong Ch(Y)$ it is enough to show that $Ch(Y)\cong H^*(Y)$. Since the diagonal is an isomorphism, the map $Ch(Y)\to H^*(Y)$ is surjective. I do not see how to prove injectiveness though.