Try https proxy for websocket connection.

net/socket_stream/socket_stream.h

 // Copyright (c) 2009 The Chromium Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.
 
 #ifndef NET_SOCKET_STREAM_SOCKET_STREAM_H_
 #define NET_SOCKET_STREAM_SOCKET_STREAM_H_
 
 #include <deque>
 #include <map>
 #include <string>
@@ skipping 252 matching lines @@
 
   typedef std::map<const void*, linked_ptr<UserData> > UserDataMap;
   UserDataMap user_data_;
 
   State next_state_;
   scoped_refptr<HostResolver> host_resolver_;
   ClientSocketFactory* factory_;
 
   ProxyMode proxy_mode_;
 
+  GURL proxy_url_;
   ProxyService::PacRequest* pac_request_;
   ProxyInfo proxy_info_;
 
   HttpAuthCache auth_cache_;
   scoped_refptr<HttpAuthHandler> auth_handler_;
   HttpAuth::Identity auth_identity_;
   scoped_refptr<AuthChallengeInfo> auth_info_;
 
   scoped_refptr<RequestHeaders> tunnel_request_headers_;
   size_t tunnel_request_headers_bytes_sent_;
@@ skipping 34 matching lines @@
   scoped_ptr<SocketStreamMetrics> metrics_;
 
   RequestTracker<SocketStream>::Node request_tracker_node_;
 
   DISALLOW_COPY_AND_ASSIGN(SocketStream);
 };
 
 }  // namespace net
 
 #endif  // NET_SOCKET_STREAM_SOCKET_STREAM_H_