Update: I got a Jabra XS10 Series 2 headset yesterday. It has been working fine, including at least one period of an hour or more with an idle connection, with the Bluetooth LED indicator set to On.

A little while ago, I tried setting the BT LED indicator back to Off. Approximately 2 minutes later, the phones Bluetooth went to Disabled. I went into BT options and Enabled BT again, then turned the LED indicator back on. The headset reconnected automatically, as soon as I enabled BT again, and now (with the LED indicator back on), the headset has maintained its idle connection for another 30 minutes already.