Is there a known correlation between human voice (360Hz to 5kHz) and the frequencies above/below that?
There is a correlation and it can be learned with machine learning methods. A simple google search can give you many papers, recent research

Waveform Modeling and Generation Using Hierarchical Recurrent Neural Networks for Speech Bandwidth Extension
Zhen-Hua Ling, Yang Ai, Yu Gu, Li-Rong Dai

older paper:

Techniques for the Regeneration of Wideband Speech from Narrowband Speech
Jason A. Fuemmeler, Russell C. Hardie, William R. Gardner