@billroy: I tried replacing "GET /" with "GET /pub/WWW/", to no avail. Also, Heroku does not support HTTP 1.1.
@PaulS: My bad, I left that bit out. Here you are:
char serverName[] = "ruby-coffee-maker.herokuapp.com";
I was a bit concerned that "-" was an illegal character, but changing it on the server side to not have hyphens yielded a net gain of nada.