Can you please make sure that those image exist in /content/dam/geometrixx in author. You can get Image path from publish instance.
Yes, they exists but in the path /content/dam/geometrixx/banners/ . All 4 images.
I also noticed that the events pages (the destination pages declared in Fixed List carousel component) can't keep the images stored in the Image tab in Page Properties.
I Drag and Drop the images from DAM, everthing looks Ok but if you check page properties image tab again it is blank.
Do you see any JS error ? Also do you see all banner under http://HOST:PORT/siteadmin#/content/campaigns/geometrixx-outdoors/banner with right template ? Or when you right click on banner do you see Compaign path set to right path (that is /content/campaigns/geometrixx-outdoors/banner) Other thing I can think of is, May be segmentation information missing (But in that case it should take default segment Image).
Banner images are all in /content/dam/geometrixx/banners and pages in the Carousel fixed list points to default Geometrixx demo /content/geometrixx/en/events/.
No Campaigns or segments involved. No JS problem/alert showed.
Carousel works fine in the publish instance. Problem only occurs in Author instance.
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One of the possibility is you could be missing OR disabled the bundle "Apache Sling Scripting Java Support" 
Make sure it exists & activated. Please verify the online tool launched to sanity check