I have communicated with Litespeed support and they responded as follows and resolved the issue for me:
“this happens because your site theme is using POST /wp-json/hivepress/v1/users/login/ to do the login action , unlike normal wordpress site that doing the POST to the page itself
and our plugin will set a cookie _lscache_vary to identify and vary the login users , however , because the login happens takes place on that wp-json URI, so such cookie didn’t set”
“ I added the wordpress’s default login cookie to Cache → Advance → Vary Cookies , as my screenshot”.