The issue seems isolated to the homepage only as the carousel works properly on a test page.
Let’s debug further.
Can you try disabling ALL plugins except WPSP and WPSP Pro to see if a plugin is messing in conflict with the it?
If the issue still occurs, can you try deactivating the child theme and see if that’s where the conflict is?
Lastly, if the issue still occurs, can you try temporarily changing to a default WordPress theme and see if anything changes?