Purging private cache on wp_ajax

`header("X-LiteSpeed-Purge: private, *");`
This method was working in wp_ajax but stopped working for a while ago.

`do_action( 'litespeed_purge_private_all' );`
I tried to use this action instead, but it didn't worked in wp_ajax.

Example action;
```
add_action( 'wp_ajax_custom_action', 'custom_action' );
function custom_action() {
do_action( 'litespeed_purge_private_all' );
//do some stuff
wp_die();
}
```

Start by reproducing the provided custom wp_ajax action and compare the X-LiteSpeed-Purge header with the litespeed_purge_private_all action. Trace how private-cache purging is handled during AJAX requests; done means the example action reliably purges the private cache again.

Written by the indexing model from the issue text.