Exaustion is not covered by the effect.
The key portions of the effect of the curse are:
the creature's hit point maximum is reduced by the Soulmonger
other life-draining effect, such as (...)
This [not defining what life-drain is] might be an undersight from the developers. They are creating an umbrella definition for "life draining" (reminiscent of the old AD&D "energy drain" or 3.5 "negative levels" mechanics) without explaining it fully. But from the list of examples, we can know what it means: those necrotic damage types that cannot be healed until a certain condition is met, usually caused from undead attacks.
In the line of ruling from the sage advice, the effects do what they say, not what they don't say.
I'm applying two 5e premises:
First they go into the general "HP reduction". Then they specify what kinds of HP reduction are included in the effect. Most of the rules discussions about 5e are caused by people wanting to take one piece of writing beyond what is written. There is no "hidden subtext".
[Making a Jeremy Crawford voice impersonation]: "If the curse included the HP reduction of exaustion, it would say so"
So only in the two cases above, will the curse prevent the HP from coming back.
The HP reduction from exaustion will be lifted when the exaustion level that caused it is removed.
Also, the HP max reduction is not permanent. When (and if) the curse is lifted, people can heal. More info in the primer pdf. It is mostly for AL play, but the curse's rulings work for table play too.
Another possible way out maybe is to go die in another plane of existance, as this sage advice hints at.