Your settings should theoretically work. To rule out GC being the 
issue, you can set the gc_probability temporarily to 0, and/or run 
your own garbage collection.

Please use the mailing lists to ask for further support since the 
default setup should work as is.

http://www.horde.org/mail/ contains a list of all available mailing lists.

This makes sense to me in theory, but we are definitely still having 
Horde sessions timeout. We're using the default 
session.gc_maxlifetimesession.gc_maxlifetime = 1440, 
$conf['sessionhandler']['type'] = 'Builtin', and 
$_prefs['menu_refresh_time']['value'] = 300. With these settings, I 
would expect to effectively have unlimited Horde sessions as the 
menu_refresh_time is shorter than the gx_maxlifetime.

As a temporary experiment, I was considering setting "php_value 
session.gc_maxlifetimesession.gc_maxlifetime 86400" for the Horde 
vhost in my Apache httpd.conf file, but I wanted to make sure that I 
wasn't missing something first.

First of all, it's hard to determine the exact end of a session, 
because it depends on many factors, including statistically ones like 
the garbage collector. Unless you have configured a finite session 
timeout in Horde.
Beside that, the session is already being refreshed as long as the 
user has Horde open, since the menu is updated regularly, effectively 
sending a ping to the session handler.

I would like a user-friendly mechanism to warn users that their 
session is about to expire. Currently, using the default PHP session 
handler, the session can be terminated and the customer could be in 
the middle of composing an email or other activities that will then 
fail. Maybe a popup or something similar that the customer can click 
to extend their session? I.e. "Are you still there?"