Hi,
It is because the sticky menu makes a DOM copy of the search HTML and places it into itself. This results in invalid HTML and loss of attached event handlers, as copying HTML elements does not copy the event handlers as well.
Unfortunately this is not possible to resolve with any custom code, as making duplicate elements of same IDs and names is not valid, and there is no way around it. It is not related to ajax search pro, unfortunately, if you would place any script powered element into the sticky menu, it would very likely fail as well.
Best,Ernest Marcinko
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