Let $D\subset X$ be a smooth divisor on a smooth variety over $\mathbb{C}$.

Then we have Gysin maps in étale cohomology $H^2(X\backslash D,\mu_2)\rightarrow H^1(D,\mathbb{Z}/2)$ as well as $H^2(k(X),\mu_2)\rightarrow H^1(k(D),\mathbb{Z}/2)$ for the function fields.

$\textbf{Question:}$ Are these two Gysin maps related by a commutative diagramm of the form

$\require{AMScd} \begin{CD} H^2(X\backslash D,\mu_2) @>{Gysin}>> H^1(D,\mathbb{Z}/2)\\ @V{p}VV @VV{q}V \\ H^2(k(X),\mu_2) @>{Gysin}>> H^1(k(D),\mathbb{Z}/2) \end{CD}$

with $p$ and $q$ injective, where both maps are induced by the inclusion of the generic point of $X\backslash D$ and $D$ resp.

$\textbf{Reference request:}$ Using Jason Starr's comment, this would follow from the functoriality of the Gysin sequence in the pair $(X,D)$. I have seen this mentioned in some articles without proof. Does anybody have a reference (some book / article) where this fact is proven?

Dix Exposeswhere this is discussed is Section 6 of "Le groupe de Brauer III" on p. 133. There is a further reference to SGA 4, Expos'es V and VIII, particularly VIII.6. In Milne'sLectures on Etale Cohomology, see the proof of Theorem 16.1. $\endgroup$