If $k$ is a perfect field (not necessarily separably closed), $f : X\to Y$ a proper birational map of smooth projective $k$-varieties, $\ell$ a prime invertible in $k$, is the induced map, for **some** $j$

$$f^* : H^j_{ét}(Y, \mu_{\ell^n}^{\otimes i})\to H^j_{ét}(X, \mu_{\ell^n}^{\otimes i})$$ an isomorphism, for every $n\ge 1$?

For example, if $k$ is of characteristic zero and $i=1$, $j=2$, we have

$$0\to \text{CH}^1(X)/\ell^n\to H^2(X,\mu_{\ell^n})\to \text{Br}(X)[\ell^n]\to 0$$

and left and right sides are birational invariants.