I'm actually not sure what will happen to White and the resulting consequences of him will be but I suspect it will be a more a Mexican standoff over white that determines how things in the US got so crazy.This isn't like a murder or something, this is full-level treason on the highest levels. It's not getting swept under the rug even if the GOP wanted it to
It benefits the soviet union that White dies as much a martyr as possible and preferably in a way leaves political confusion and chaos for years if he is at risk of being discovered. That way the newly allied bits of his term (Jews, blacks hoping he can stop segregation, women, people benefitting his socialist policies) bitterly oppose changing and cause a temporary deadlock while many of White administration are ''purged'' to not leave evidence behind and also push any last minute things they can in the US and abroad.
It benefits the shadow clique to have White captured alive and ''interrogated'' to admit to many things both that way they can set the narrative and the fallout while also showing they are not lunatics but obey the US legal system.
It benefits the fringe right wing/conspiracy extremists that are a bit attached to the shadow clique but are their thing clearly convinced white is the devil and killing/removing him from power is first priority damn what happens next.
I admit I do suspect the shadow clique will ultimately at getting a good transition when white dies and as a result the Jewish conspiracy and others become state dogma while the soviet union does everything it can to throw oil on the fire both to destabilize the US and also by time for their projects.