Computer keeps 'loosing' one of the two attached displays.

My graphics adapter is an 'ATI RADEON HD5770'. I have two displays attached, a 23" Cinema HD, and a 20" Cinema. It has worked OK for a year or so and now I have a problem where the computer 'looses' the 20" display (assgined as monitor 2) at power-up or when waking up from a 'power save, display sleep'. At first I thought the problem was the display itself, but if I swap displays (swap the ports the displays are plugged into), the 20" works OK and the 23" goes dark (into stand-by). Doing a 'restart' doesn't help, however, if I unplug the computer and plug it back in again, the problem will go away, and both monitors will start working again. After a few more daily power-ups, it will fail again, and unpluging and plugging in the computer will 'fix' it for a few more power-ups, then it fails. To be clear, when I say 'fail', I mean the computer goes back into a single monitor mode.

I discovered this morning that there is one other way to 'fix' it (i.e. getting it back into a dual monitor mode). Normally I have each monitor connected via a mini display port adapter. I can get things working again by disconnecting the 20' monitor from the mini display port and plugging it into the 5770's DVI connector. It will work this way through a few more power-ups, then fail again, this time being 'fixed' by moving it back to the mini display port. Sorry, I know how confusing all this may sound, but I don't know how to make it any clearer. My gut feeling is the problem is with the 20" monitor... it fail the power-up handshake and the computer assumes only one monitor is attaced. Power-cycling the computer or swapping mini display port and DVI port initiates another handshake session, the 20" reports in this time and the computer goes back into a dual monitor mode. The displays are rather old and no longer under warranty. I don't want to go to the expense of replacing the 20" until I'm absolutely sure it's the problem. I could live with it the way it is, but unfortunately it keeps getting worse (i.e. it is failing more frequently). Any ideas?