Search header tag inside page header

[laurentpayot]

Beercss search requires a header tag, docked or fullscreen.

That’s a problem because if you want to put a search input in your app bar you will end up with the header tag of the search inside the header tag of the app bar (the page header). You should not do that according the the HTML standard specifications for the header tag:

Content model:
Flow content, but with no header or footer element descendants.

An example here: https://codepen.io/laurentpayot/pen/JjeZLJx

So is the header tag really necessary for Beercss search?
Could we replace it by a section tag or an aside tag? Or even a div tag with a special class?