Private cache fails

Hi.

I am having hard time to make private cache work. The site is running Learndash, Woo, some heavy stuff and private cache works only for Admin users, but never for others, like customer users. I tried resetting plugin settings, without success. Cache for logged in users is on, but it does not help.
Here is an example of what a logged in customer user gets:
x-litespeed-cache: miss
x-litespeed-cache-control: private,max-age=1800
x-litespeed-tag: fc6_tag_priv,fc6_ESI.storefront-cart-header,public:fc6_HTTP.200,public:fc6_front,public:fc6_URL.6666cd76f96956469e7be39d750cc7d9,public:fc6_F,public:fc6_Po.8247,public:fc6_PGS,public:fc6_

Appreciate any help
Thank you,
Alex