Istio Sidecar Connection Pooling?

I've been reading documentation up and down all day and I can't seem to get this to work.

I have an unruly application that opens a connection for every HTTP request. I would like to improve performance by forcing HTTP multiplexing over long lived TCP connections.

I tried making a ServiceEntry and a DestinationRule, but I didn't see that have any effect. I still saw a large number of TCP connections made.

I figured Istio would pool the connections via maxRequestsPerConnection from DestinationRule. Is that wrong?

I imagined I needed:

1. A ServiceEntry to make known the external service.
2. A DestinationRule to govern how the service is accessed.

In general though I would just like the Envoy side car to just pool and multiplex all egress connections for this service no matter the destination.

Destination Rule

apiVersion: networking.istio.io/v1alpha3
kind: DestinationRule
metadata:
  name: api
spec:
  host: google.com
  trafficPolicy:
    connectionPool:
      tcp:
        maxConnections: 100
        connectTimeout: 30ms
        tcpKeepalive:
          time: 7200s
          interval: 75s

Service Entry

apiVersion: networking.istio.io/v1alpha3
kind: ServiceEntry
metadata:
  name: api
spec:
  hosts:
  - google.com
  location: MESH_EXTERNAL
  ports:
  - number: 443
    name: https
    protocol: TLS
  resolution: DNS

istioctl -i istio proxy-config cluster api.namespace
...
google.com    443       -          outbound      STRICT_DNS       api.namespace
...

istioctl -i istio proxy-config cluster api.namespace --fqdn google.com -o json

        "circuitBreakers": {
            "thresholds": [
                {
                    "maxConnections": 100,
                    "maxPendingRequests": 4294967295,
                    "maxRequests": 4294967295,
                    "maxRetries": 4294967295,
                    "trackRemaining": true
                }
            ]
        },


        "upstreamConnectionOptions": {
            "tcpKeepalive": {
                "keepaliveTime": 7200,
                "keepaliveInterval": 75
            }
        },

OK, so I think what I discovered above in my question is that maxConnections does not cause the Envoy Proxy to make a pool with 100 connections. Instead this sets a circuit breaking rule that prevents the application from opening more connections.

Perhaps I need to use an HTTP Connection pool?