Paid Search bias - Nested model

[ar-asur]

Using nested models is often prescribed for mitigating search spend being endogenous variable. I read this Google paper(Bias Correction For Paid Search), It is mentioned we can use google query volume(V) to correct the bias. In the implementation part, it is said that incorporating google query volume as an additional control variable into traditional media mix models will still have the estimates to be biased. One possible approach mentioned is ,

Model 1: Predict sales as a function of search spend along with google search volume variables. Model 2: Predict sales as a function of the search spend + other media channels and control variables. We need to fix the impact of search spend as is from model 1.

I am not sure how to keep the impact of the search spend fixed in the model 2 in this case. Should it be done via adjusting the prior for the search spend

Thanks!