Some pseudocode, if you will.
Nothing final, just getting thoughts out...
(All Normal Stuff Happening)
When move 100 (arbitrary value indicating endgame) comes around and expected score percentage is 75% (again arbitrary). Another condition could also be that the cp eval is very high or low (abs>100 or 150 or whatever - these values are again arbitrary):
Board position before white moves at Move 100 stored, perhaps in FEN format
Move 100: white moves and cp (centipawn) eval of white stored in array[0]. Black moves.
Move 101: white moves and cp eval of white again stored but this time in in array1]. Black moves.
Move 102: white moves and cp eval of white again stored but this time in in array[2]. Black moves.
If array[0]==array[1]==array[2] (or perhaps equivalent enough within a small enough margin), then
restart at the FEN position obtained at Move 100
Let normal MCTS Search happen (from that same position, of course)
Note that this means the same move could perhaps be chosen, in which case this would of course repeat
Eventually, the move that was chosen would make a significant difference, whether as a "good" move (cp eval increases) or a "bad" move (cp eval decreases)
Of course, one way to also do this, would be every three or five or ten or however many moves, see if the cp eval is high enough that it would be indicative of a win but has still not changed BUT as others have pointed out above, at the beginning of the game, this would be somewhat silly as the games are generally drawn and constant for a while anyways. Having said that, it might be more viable past a certain point (ie, every three moves past move 75 or 100 or whatever).
Note that the "pseudocode" I wrote before has one major difference from what I have written before. That is, rather than store the played move and making sure it is not played again, we simply default back to the standard MCTS.
With enough of these sort of games occurring, eventually when the weights are calculated or updated every four hours on Error323's computer, there should be a significant difference in the ability of the network to convert its advantage to a win. We should also see considerably quicker games as rather than proceed down paths that lead nowhere, LCZ will be able to choose more efficient routes, allowing more games to be played in the same amount of time given the same amount of processing power (eventually, although it will likely take longer at first).
As a final point, the amount of human "interference" in this would still be negligible as the positions that Leela would reset at would not be forcefully given to her from the outside (i.e. by people). They would be whatever board positions LCZ had already generated and arrived at of its accord.