The White Army was a very loose coalition of often radically opposed groups, held together mostly by their shared hostility to the Bolshevik government. Tsarists, liberals, ethnic nationalists, Cossack traditionalists,
deranged messianic warlords. If the Bolsheviks are gone, these factions turn on each other and the conflict just rolls on, fighting each other. The tensions which toppled the old regime had erupted into open conflict before Lenin left Switzerland, and wouldn't simply dissipate if he left Moscow.
It would probably turn out like Mexico, dragging on for a decade or more, and the factions and personalities who end the conflict are totally different from those who started it.
I think that you probably end up with some conservative nationalist regime, but what that regime looks like or calls itself, and what it's borders turn out to be, is anyone's guess, and would probably have been as much a surprise to the participants as it would be for us.