cache problems on wpengine

I have installed your searchiq plugin on live and staging site, exactly same configuration, different API keys of course, and the both work ok for live search result, but on live site, that it’s cached, the search result page shows no results.. on browser console it also does not seens to load the scripts loading on uncached version..

Here is the comments from hosting staff:

I’ve excluded the entire plugin from cache, along with the /search page and any page with the ?q= argument in it.
As the plugin works on staging (completely uncached) but not live (cached) it is almost assuredly a caching conflict.
It does seem this plugin needs something more excluded from caching.
Can you reach out the developer and request any arguments, pages or cookies that need to be excluded for this to work?

..so can you please help with that?