I resolved my problem!
I used HAProxy instead of Lighttpd mod_proxy as specified in this question
Here is my conf file (amend <...>
per your configuration):
# this config needs haproxy-1.1.28 or haproxy-1.2.1
global
log 127.0.0.1 local0
log 127.0.0.1 local1 notice
maxconn 4096
uid 99
gid 99
daemon
defaults
log global
mode http
option httplog
option dontlognull
retries 3
option http-use-proxy-header
option redispatch
option http-server-close
maxconn 2000
contimeout 5000
clitimeout 50000
srvtimeout 50000
frontend public
bind *:80
acl is_example hdr_end(host) -i <URL.toyourwebsite.com>
acl is_websocket hdr(Upgrade) -i WebSocket
acl is_websocket path_beg -i /websockets
use_backend ws if is_websocket is_example
default_backend www
backend ws
balance roundrobin
option forwardfor # This sets X-Forwarded-For
timeout queue 5000
timeout server 86400000
timeout connect 86400000
server apiserver localhost:<PORT> weight 1 maxconn 1024 check
And I made Lighttpd listened to the 8080 port (otherwise HAProxy wouldn't start).