What are some PHP and Apache settings that could cause empty response body from a Sugar API endpoint?

I'm working with a customer of ours (a Sugar customer) that recently upgraded to PHP 7. They use our Zendesk/Sugar integration. Our integration worked perfectly before, but as soon as the php upgrade went through, calls from Zendesk to our Sugar endpoint stopped working.

When we make a call to our endpoint https://{sugar_instance}/rest/v10/FanaticalZendesk/app Zendesk receives a 200 response code but the request body is completely empty. All of the logs in Sugar look perfect. We log the expected response in the Sugar logs and everything is there. It's just when apache actually sends the response back to Zendesk, something isn't right. I'm guessing there is a PHP or Apache setting that isn't allowing the outgoing response for some reason. 

I've tested other Sugar instances with their Zendesk instance and it works just fine, so I'm pretty sure the issue is isolated to their specific Sugar web server.

Can you think of any settings that would cause something like this?

Any other ideas? I'm completely stuck on this one. Thanks!