Live Preview has gone! A provider/server related thing?

Hi, I manage about 10 WordPress websites (latest upgrade, different
premium themes/Twenty Twelve) with hosting provider X, and about 8 sites with same themes with other providers Y and Z.

Few weeks ago I discovered that in all 10 sites hosted by X the Live Preview in Customizaton was gone: blank preview box. I can’t think of a specific cause. I still can customize specifications, but I can’t see them changing right away; box stays blank.

The other sites hosted by providers Y and Z don’t have any problems.
So the missing Live Preview seems a provider/server-related issue.
But provider X will not give any script related service.

I love the Live Preview feature, I can live without it but this is not what the gods of WordPress had in mind. I really would like to know what is happening. I’ve searched the forums, but I found outdated comments, no real answers.
I use fresh installations, latest WP and theme updates, deactivated all plugins, … you name it. I hope one of you guys has an idea. Thanks in advance for ideas.