I can't imagine why someone would follow the southern possibility, it leads no-where within fuel range. Surely it must have gone North?
I'm not sure why you think this. Perhaps you can explain your thought process.
With the supplied information, I think the southern route is still plausible. You can't discount it until there is data to support removing the possibility.
1. You have data supporting the poss. of this route.
2. You have an AC that potentially flew to fuel exhaustion. This adds another element to consider. The last data point when the AC could have potentially run out of fuel, is telling. If fuel exhaustion could be ruled out, it would be one less variable.
3. You have no SAR team or anyone anywhere spotting anything from the AC.
4. No pings from any transmitter (ELT, FDR, etc.)
5. You have very deep water.