[exim] headers_rewrite not working

Hi.

I am trying to rewrite the *from* and *sender* headers to remove the
hostname, leaving just the domain name, only on outbound mail.

Here are the related parts of my configuration.

begin routers

  mail_relay:
      driver = manualroute
      domains = !+local_domains
      transport = remote_smtp
      route_list = * our.smtpserver.com

begin transports

  remote_smtp:
      driver = smtp
      headers_rewrite = *@*.mydomain.org $1@??? fs

So far as I can tell, this is exactly how the docs say to do it,
but the *headers_rewrite* rule is having no effect. If I put

*@*.mydomain.org $1@??? fs

in the global "begin rewrite" section, the *exim -brw* test indicates
that my rewrite syntax is correct.

  $ exim -brw me@??? 
    sender: me@???
      from: me@???
        to: me@???
        cc: me@???
       bcc: me@???
  reply-to: me@???
  env-from: me@???
    env-to: me@???

Why would the headers_rewrite rule have no effect? It is not removing
the hostname.

Any help would be appreciated.

I am running exim-4.67 on BSD.