I've just noticed that you're on e Windows server - Unfortunately I can only answer this from a Linux point of view... so ignore my earler answer about it being possible as that's Linux. It may be on Windows, but if the techs say "no" then that's the only advice I've got.
However, I've just checked the details for Plesk for Windows and you CAN have certificates per domain.
I've also checked on a Windows account and you CAN go to
https://yourdomain.com:8443 and log in that way. So I really don't see why you can't do it, unless some quirk of Plesk will only provide the generic certificate...
You can create yourself a certificate (do it for free - just don't use a trusting authority) and upload and point your browser at
https://yourdomain.com:8443 and see if it works. It'll cost you nothing except a few minutes. I'm sure it'll work
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The generic certificate this is slightly historic and based on user's comments and feedback and debate at the time HostNexus started out. It took about a year for it to be settled on but there was a lot of discussion at the time. I can't remember the ins and outs, though.