Honestly, i think you've lost me!
You said the problem is "unsolvable" without "load or line phase info" but you question me talking on the same issue? Load info will be manifested on the load current magnitude and the load angle. The scenarios I presented were all about how the load current and their respective angles point into a resolution of the mismatches.
Anyway, I think this is a dead horse already. We might as well stop beating it.
You assigned the phase angles for the line currents as being balanced
they are not, you assumed the imbalance is only based on magnitude not phase
ie only R no X, which likely not the case
niether are the phase voltages balanced which you summed to zero
you set up the system with a predetermined outcome based on the data constraints YOU assigned
not the actual ones for the system (which are unknown)