Speaking of the design of the MP2658, it looks like the recommended battery range here for this supercapacitor charger IC is as follows, ranging from 2V - 31V.
In addition to this, here is the specified pre-charge and trickle charge voltage ranges.
This sort of corroborates with the 1.5V - 2V range in which you see charging. Would this external pre-charge to 2V be something that can be implemented in your application?