One of the most interesting articles I ever read was in SciAm, arguing (from a thought experiment, Maxwell's Demon [1]) that information needs some minimum energy to be erased [2] (=reset to a low entropy state). Thus, there's definitely a connection between energy and information. Given the known connection between energy and mass, this would establish the final leg of the triangle.

[1] https://en.wikipedia.org/wiki/Maxwell's_demon

[2] Charles H. Bennett Demons, Engines and the Second Law from 1987.

https://www.scientificamerican.com/article/demons-engines-an...

https://ecee.colorado.edu/~ecen4555/SourceMaterial/DemonsEng...