When sending a request you can pass the following keepalive settings which will keep the connection open:

local http = require "resty.http"
local httpc = http.new()
httpc:connect("127.0.0.1", 9081)
local response, err = httpc:request({
  path = "/proxy" .. ngx.var.request_uri, 
  method = "HEAD",
  headers = ngx.req.get_headers(),
  keepalive_timeout = 60,
  keepalive_pool = 10,
})