I highly doubt that. Here is my thought.
The H-III's main engine is LE-9. The throttle range is down to 63%. The core stage can have 2 or 3 LE-9s. Only the variant of 3 is possible to be reused. When landing, two engines will be shutdown, that leaves one engine, its full thrust being throttled down to its minimum 63% will give the minimum thrust at 0.63/3=21% of the liftoff thrust.
We know that rocket using LOX/Kerosene like Falcon 9 need to throttle down to 10% to make around 1:1 thrust/weight ratio to enable powered landing. The last moment is like hovering over. If the thrust is more than that, the rocket will not land but go up instead. To couple with larger than 1 ratio, the engine has to be shut down when the rocket is just a meter or so above ground, essentially smashing the rocket on ground. This is doable as SpaceX has done it in early days with higher chance of failure of course. The 10% figure is based on the fact that the dry weight of a rocket (tank almost empty) is about 10% of liftoff mass. This is determined by the type of fuel Kerosene that determines the size of the tank and additional structures for it. LH2 is much light but its tank is much bulkier for the same payload, so the key question is what is the dry mass to liftoff mass ratio? If it is around 21%, then fine. But I seriously doubt that Japan is that bad in making a light weight tank for LH2 even if we take into consideration of the bulkiness of LH2 tank. Take Saturn V's second stage as a reference, its dry mass was 9.7% even better than 10%. Japan should not be much worse than that considering Japan has been working on LH2 rocket for decades and it has a good material manufacturing foundation. So its ratio could be just above 10%, still too light for the minimum thrust of 21%. To reach around 10%, LE-9 has to be further throttled down half to 30%. A LH2 engine of this thrust class (>100 tonne) to 30% is nothing easy, even the most experienced and still the most advanced countries in LH2 engine (U.S. and Russia) have not done it, and probably not able to do it in the near future.
In conclusion, LE-9 and H-III in their current design specifications are very unlikely to be converted to being reusable, almost impossible. It would be another rocket and engine if they choose to do so.
Where are you getting the throttle range figures for the LE-9?
Expander cycle engines are typically able to throttle down much further than a gas generator engine.
That is one reason why the DC-X demonstrator used an RL-10 as the engine.
Look at the throttle range of the LE-5B which has known figures for example. It also uses expander bleed cycle.
Expander cycle engines are also typically capable of multiple restarts mid-flight which also makes them suitable for reusability.
The US uses the RL-10 for this. You can insert satellites into multiple orbits for satellite constellations with that upper stage rocket engine.
Expander cycle is also amenable to use with other cryogenic fuels. For example the RL-10 was designed as a LOX/LH2 engine but it was bench tested with higher density LOX/LCH4 (methane) and it worked fine.
With regards to the tank mass structures I wouldn't be surprised if the dry mass fraction is much higher than that of the Apollo V second stage. I think the H-III uses tanks derived from those of the H-II to reduce R&D costs. AFAIK they didn't lighten up the tanks at all.
Ariane 5 is another rocket that uses tank manufacturing that was obsolete at the time it was designed even.
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