Actually - just tried your website and it doesn't work for me. I tried curl -H "Content-Type: application/json" -d '{"message":"hello world"}' https://8ef216dd46.endpoints.dev and nothing happened. Maybe check your logs.
Are you running behind a load balancer on AWS? You might have to increase the idle connection timeout from the default (60 seconds). I use the max 4000s on hopps.io. Sending keep alive's to prevent the socket connection from closing also helps. Not sure if that's your issue, but worth looking into...
hmm - I can see a few requests have been received & stored for that endpoint.. It could be that your client killed the connection to the websocket api - does refreshing the page show the requests?