Probably not. Cryogenic liquids like liquid H2 are typically stored in dewers, which use a vacuum as the primary insulation, along with silvered surfaces and layers of mylar film to reduce radiation losses. We've been able to make dewers with boil off rates of 1% per day or better for decades. That's more than low enough that you don't need any refrigeration at all for an aircraft application where the flight just lasts a few hours anyway.
Most likely they'll actually use less effective insulation to save weight. Fortunately for them, cryogenic liquids are commonly used in rockets, so there's already lots of know-how and technology out there for making lightweight cryogenic insulation.
In electric, heat is the enemy and inefficient.
In combustion, heat is a key component of the thermodynamic efficiency, unavoidable, and effectively only limited by material tolerances.
(I didn't major in physics, so someone keep me honest on the ceiling / floor efficiency equations for combustion engines?)
Combustion engine's are also limited by Carnot cycle engine efficiency - about 64% theoretically, no more then 50% in the real world. So you're still going to wind up having to dump progressively more heat from a fixed surface area.
Carnot cycle was what I was grasping for.
Original point being that in a thermal engine, increasing your temperature gradient by increasing your high-temperature side (while holding exhaust constant) leads to more potential power.
Or IOW, a thermal engine that doesn't burn hot somewhere isn't going to produce much power.
(Electric being a completely different ball of wax)
Rockets do this today, as indeed they have done from the earliest days, sending liquified gases through channels in the engine bell and combustion chamber before burning, keeping them both from vaporizing.
Here, the very high efficiency of electric-driven turbines is offset by the lower efficiency of fuel-cell generation, for what is expected to be a substantial net gain over burning H2 in internal combustion turbines.
The latter are the obvious first stage of deployment, taking advantage of LH2's extreme energy-to-weight ratio for reducing the takeoff fuel load by as much as 5:1. This form will probably be fielded in existing airframes, displacing some cargo space with in-hull LH2 tanks.
This is regularly done by LNG tanker ships. Evaporating LNG is simply routed to the engines.