"We have tried disabling mu-plugins. However, the issue persists. The best option is to disable all the plugins except WPML and then switch the theme to default to see if that issue persists. In case you want us to check it for you, please provide us the authorization to disable themes and plugins."
So that is the next step. I don't think it will matter, but I'm letting them go down that path.
I've tried applying the workaround on your website but it didn't effect anything. But anyway, our 2nd Tier Support has already escalated this to our devs and they will review and investigate this further.
I am currently fighting with Nexcess to look into it. They said they don't support the plugin and said to check with you guys and my own developers. I called them on the phone, spent 30 minutes with them, and finally convinced them it is their problem and they need to look into it further.
If we don't get this resolved, I'll either have to move hosting or go with another solution. Both are definitely unappealing.
Sorry, I cannot give any advice for this since this is related to your hosting. Maybe you can test it on a temporary 3rd party server and compare the results?