Cannot update network due to a specific site

Hello everyone!
Our main WPMU has an error when we try to update the network (after installing WordPress 5.0.1).

Attention! Un problème est survenu lors de la mise à jour de https://www.agquebec.com. Votre serveur peut ne pas être en mesure de se connecter à certains sites qui y sont installés. Message d'erreur : cURL error 35: error:14077410:SSL routines:SSL23_GET_SERVER_HELLO:sslv3 alert handshake failure.

We had this error on a previous website due to WPML, but this time we can't find why it's happening.
- We don't have more details in debug.log
- The SSL Certificate seems to be fully valid
- It's not the only site with this factor, but its DNS are external (nameservers pointing elsewhere)

Anyone has an idea of something we can try or improve before we try to deactivate the non-network plugins one by one?

We'd like to learn what this network update is for exactly and what could make it fail like this. :slight_smile: