[Plugin: WordPress HTTPS] HTTPS sessions

Hello, I’m using the plugin WordPress HTTPS. Is it supposed to maintain the session when the user switches from HTTPS to HTTP? I explain my problem in details.

I’ve a shared certificate, so, when the user does the login, he’s redirect to a page https://path.to.shared.certificate/wordpress/wp-login.php
The login works, but, when he clicks on the link to visit site, he’s redirect to https://www.mydomain.com, but it looks like the session is lost, because he’s not logged in.

Is there a way to fix that? Thank you so much!