Looks like the updates are block too. In fact, the update notice does not even show and one should for ROT13. It’s still the current version, I simply changed the version number so the update should be shown.
I see you use BackupBuddy (BB) too. But it is not the most recent version. A newer version is available yet there is no update notice for it. Have you been able to update BB on the site in the past? I ask because it seems updates are being blocked for that too. The BackupBuddy Settings : License admin page says the status is active but that could simply be a cached response.
There is also seems to be an update for The Events Calendar PRO which is not being displayed. Perhaps it is being blocked too. On the Settings : Licenses admin page it reports the error “Sorry, key validation server is not available.”
Checking the other commercial plugins, I’m unable to tell if they are the current version or not since they do not seem to tbe display the current version on their sites doc pages.
My guess is that this issue is not affecting only Connections but all the other commercial plugins too. Honestly, though, just a guess at this point as more testing and verification would be in order.
Now, since it seems the core WordPress update seems to be working fine when it is coming from the wp.org sites my best guess is the host is applying a whitelist of allowed sites. Since they will not open cURL for you, ask if they will at least add the following URL to their whitelist.