I am also searching for same. I figured that logged in user can call: WP_Optimize_Page_Cache_Preloader::instance()->run('manual');
But from cronjob script is not executed as it need.
It also works with bash script, but it’s not good solution for me:
#!/usr/bin/env bash
site="https://www.web.com/"
login_address="$site/wp-login.php"
log="user"
pwd="pass"
cookies="/tmp/cookies.txt"
agent="Mozilla/5.0"
# authenticate and save cookies
wget \
--user-agent="$agent" \
--save-cookies $cookies \
--keep-session-cookies \
--delete-after \
--post-data="log=$log&pwd=$pwd&testcookie=1" \
"$login_address"
# access home page with authenticated cookies
wget \
--user-agent="$agent" \
--load-cookies $cookies \
"www.web.com/test.php"