There is a ticket associated with this third party minicart module issue. Our developer gave the following advise for other's interest:
normally, need render minicart module as ESI module, and set ESI Module Cache Type to be private.
If found some bug after render it to ESI, can try set ESI Module Type to be Advanced.
If still not work, try put LiteSpeed Cache Plugin order before minicart related plugins.
If still not work, please log a ticket by sending email to
support@litespeedtech.com and provide admin panel access to do more investigation.