The 379s on the mml was suggested in a Route Study. But I can't remember the exact wording. Could be units such as the class 379.
The East Midlands route study (current) and the East Midlands RUS (previously), don't mention rolling stock type in this detail, searching on 377, 379, etc finds nothing.
However the various CP5 enhancement plans do mention stock types.
In March 2014 the original had:
"
Class 377 up to 12 cars, multiple pantographs (London to Corby);"
In March 2015 the update included:
"For Corby 2017 the train service should be limited to 2 services per hour, 12-car EMU
Class 377/387 or equivalent"
The latest Sept 2016 version is:
"The Midland Main Line electrification project will include an assumption for passenger rolling stock of 12 car
class 377 or class 350 trains or their derivatives, and will carry out the infrastructure works associated with this rolling stock."
The bit 'or their derivatives' covers just about anything...