I have either a missunderstanding or cache-control has a bug or documentation is wrong. It's about cache-control directive and to purge private cache with .htaccess RewriteRule.
In LiteSpeed documentation https://docs.litespeedtech.com/lscache/devguide/controls/#cache-control there are usage...
You should better open a support ticket to get qualified answer for this question:
https://store.litespeedtech.com/store/submitticket.php?step=2&deptid=1
Check this out:
https://www.litespeedtech.com/support/forum/threads/how-to-force-a-link-to-download-a-file-instead-of-opening-it-on-the-browser.18966/#post-112308
If you are on shared hosting ask your provider if your host uses a CDN like Cloudflare or a proxy.
If nothing helps try to overwrite this header in .htaccess
header add X-Turbo-Charged-By "Name of your choice"
To me the same, because no LiteSpeed plugin caches. Each plugin is only a control panel to control and manage the cache. Caching itself will only be done by Lscache, a LSWS based cache engine.
Again, LiteSpeed Enterprise doesn't add "X-Turbo-Charged-By" header and there is no setting in WebAdmin console to disable or enable it. In WebAdmin you can only control if Server header should appear or not. This setting is what @Pong already noticed.
If you have header "X-Turbo-Charged-By"...
X-Turbo-Charged is not a default header in LiteSpeed Enterprise. Check your headers again and look for "Server" header. If you really have LiteSpeed Enterprise Server header must be "LiteSpeed", but most of the time it is Cloudflare or a nginx proxy is used.
PrestaShop has setting for mobile view, but is disabled by default, so check settings in configuration tab in LiteSpeed plugin, but purge LiteSpeed cache after you have enabled it. Purge cache button is on top of page header in the upper right corner.