Search-Bar in Header OR Navigation?

I design a online shop for a client. The searchbar is for the target group and the stakeholder very important. So I did the search in the header, on top over the navigation bar (variant 1). My client says that they want to have the search IN the navigation bar.

In my opinion, the search in the header is more focused on the users. So I want to use this option 1. However, I miss the right reasoning. Do you have arguments for one of these variants? I have unfortunately no experience reports, cases or studies found in the web.

Thanks for help.

FYI: Testing is no option at the moment.