you are spot on about the fuel/timing map.Whilst all the advise on fuel, battery, plugs etc. change are generally good and should be checked first, the earlier 749 and also the 999S all had stalling problems, especially with open exhausts. This can be traced to the ECU IAW59M. This is one of the reasons (there were other issues as well) that later bikes got the IAW5AM instead. It has among other things, different timing maps for the 749 to cure the stalling (and maybe also for the 999, but I can't confirm this). Your bike being 2006 should normally have the IAW5AM, but you can never be sure! Ducati is/was notorious for fitting whatever was on top of the parts-bin...
At any rate having a PCIII will not help the stalling, since that device only kicks in from somewhere between 3-4k rpm and doesn't modify the timing curves at all.
Solution: ditch the PCIII and flash your ECU with the Ducati Performance map from at least 2005 (they issued several flavours of those maps as well). You can find the DP maps for both ECU's. I found that even using the standard 2005 999 map in my 59M ECU on my 999S was giving better low rev running than the OEM 2003 999S map. The 2006 DP map totally changed things and now it runs like a dream! So there is hope, even if you would have the 59M ECU. If you have the 5A you should still flash it with the DP map that is specific to the later ECU. In both cases, you can sell your PCIII!