I did some tests and I could not find the reason why the automatic update is not working. I still think this is a problem of the hosting service not being able to access Themeco servers.
You can contact your hosting service provider and ask them to make sure that there is a cURL access to Themeco servers as mentioned in the troubleshooting section of the article below:
You will need to update the theme manually on your live server manually. I uploaded the zip file of the latest version in
wp-content/themes/ folder. Please use the File Manager panel of your hosting service to unzip that file. Before the unzipping, you need to rename the
x folder and make it for example
After that you will need to delete the Cornerstone plugin from the WordPress Admin, and go to X > Validation and wait for a few seconds and the new version of the plugin will be installed.
The current version of the theme is 6.2.3 and that is why the theme forces the version 3.2.3 id the Cornerstone. As the version, 3.2.4 will not compatible with the version of the theme.