Cover over ssl a mapped domain

Hi there,

Recently we bought and installed your Domain Mapping plugin to map a client domain. Here I try to summarize the scenario:

The original site: clientcompany.mysite.com
The mapped domain: store.clientcompany.com. This works.
What we want: https://store.clientcompany.com
Our certificate SSL, comodo covers store.company.com and is installed in our servers (AWS EC2).
The issue: after installing the SSL cert, I see https://store.clientcompany.com gets covered by another certificate owned by clientcompany. Furthermore, the site CSS styles aren't loaded.
Do you know if we are going the right way? Should we ask our client to deactivate their SSL cert in order to avoid conflicts with ours?
Do I need to configure something in the Domain Mapping plugin for this?

I hope you can help. Thank you,

Miguel from Gremyo