Hi Cedric,
To make sure that we are on the same page, this is happening because you use a third-party plugin that changes the original permalink structure of WordPress. This is a conflict with a third-party plugin and outside of our support scope.
As my colleague mentioned we will do our best to improve upon the possibilities of supporting such changes in the URL structure but there is no guarantee and no ETA and we do not recommend you rely upon the prospect of that for your project.
The suggestion that we have is to use the original WordPress Permalink structure or find a way to force the third party plugin to avoid changing the structure for the whole website and exempt the URLs that Cornerstone uses.
Thank you for your understanding.