When caching for Logged-In Users is enabled, Comet Cache will automatically clear the users cache whenever they login or logout (see this article for why). If you want to override this behavior, you can use an MU-Plugin.

Note: The example below assumes that users are logging in via the WordPress login page (wp-login.php) and logging out using the logout link in the WordPress Toolbar. If you have a login widget or another way of logging in and out, you may need to adjust the example below to work with your site.

Create this file and directory: wp-content/mu-plugins/cc-logged-in-users.php

Plugin Name: Disable Comet Cache Logged-In User Cache Clearing on Login/Logout
Description: Disables the automatic Comet Cache routines that clear the user cache upon login/logout when caching for Logged-In Users is enabled
Author: WebSharks, Inc.
Version: 1.0
Author URI: http://cometcache.com.

add_filter('comet_cache_invalidate_when_logged_in_postload', function ($invalidate) {
    if (empty($_REQUEST)) { // i.e., no GET/POST vars?
        return $invalidate; // Nothing to do here.
    $request_uri = !empty($_SERVER['REQUEST_URI']) ? $_SERVER['REQUEST_URI'] : '';
    $is_wp_login_php = mb_strpos($request_uri, '/wp-login.php') !== false;
    $action = !empty($_REQUEST['action']) ? $_REQUEST['action'] : ($is_wp_login_php ? 'login' : '');

    if ($is_wp_login_php && in_array($action, ['login', 'logout'], true)) {
        return $invalidate = false; // Not on this action.
    } else {
        return $invalidate; // What CC says.