i have seen something close to this before.
in my case it was while testing the 7960 with a third party Gateway. it turns out that the 7960 IP Phones don't support RTCP port monitoring, which allows the remote end to disconnect the call if no RTP from this end (7960). During my test, the RTCP timeout was set to a lower value, so the call disconnected much earlier. you might want to see if RTCP timeout is in use on the Conference phone system and what it is set to. From speacking to cisco TAC, i think the RTCP timeout is not supported (and is not queued for implementation) on the 7960.
E.