I am trying to determine if the authors of the following report missed out on an important factor or if i am the one who have missed out on something. In the following report: Bias Correction For Paid Search In Media Mix Modeling
The authors first state the back-door criteria which is one of the criteria that can be used to determine if estimates are indeed causal.
"(The back-door criterion, Pearl 2013) Given a causal diagram, a set of variables $Z$ satisfies the back-door criterion relative to an ordered pair of variables $(X, Y)$ in the diagram if: 1) no node in $Z$ is a descendant of $X$; and 2) $Z$ “blocks” every path between $X$ and $Y$ that contains an arrow into $X$."
The authors states in relation to this graph:
"Remark 5. Note that $(X1, V)$ does not satisfy the back-door criterion for $X2 → Y$, since the path $X2 ← consumer \ demand → \epsilon_2$ is not blocked. For example, $X2$ may represent social media ad spend. This suggests that the causal effect of $X2$ on sales cannot be estimated consistently by observations on $(Y, X1, X2, V)$ only."
While this is indeed correct, it leads one to believe that if we could remove $\epsilon_0$ from the outcome we would be able to accurately identify $X2$.
Notice however that $search \ queries \ V$ is a direct descendant of $X2$ thus the backdoor criterion does not hold anyway..$X1$ is also an descendant of $X2$ thus we have two violations of the first criteria in the back-door criterion.
Question: Have the authors missed out on the fact that $search \ queries \ V$ is a descendant of $X2$ or am i missing out on something and secondly, how can this be solved?