official score

Home book is official. I suggest that the base coach on the home side check the score after any inning that runs are scored. That way there is less water under the bridge before you are trying to fix a problem.

NFHS 9-2-2:
"The official scorer shall keep records as outlined in the following rules. He or she has the final authority when judgment is involved in determining whether a batter-runner’s advance to first base is the result of an error or of a
base hit. The scorebook of the home team shall be the official scorebook, unless the umpire-in-chief rules otherwise."

But the home book still needs to stand up to examination. There has to be a way to account for every batter and runner, either through scoring, left on base, or out. In your situation the runner on second either must have scored or have been out. The unpire does not need to know the score, but he is responsible to know the number of outs. So it should have been possible to match up the outs, and identify what happened to R2.

Frankly, I haven't seen this kind of disagreement since 9 year old Little League.

This happened to me in a 12U club game. It was mother's day, second game of the morning and we were one run away from run ruling the other team. Their scorekeeper, a mother, tried to argue that we had run ruled them so she and her fellow mom's could make brunch by 1pm.

I was kind of caught off guard as we were suddenly 'agruing' against our own teams. I told her she could have it if she wanted it, the umpire came over and asked if we figured it out, I pointed to her and she said "Ball game blue". She had the official book so that was that. She smiled and suggested I take one of my son's outs and score it a HR, then they packed up and left.

We saved an inning of pitching and they got to eat brunch.

Other times this has happened, when one side has a different score, the umpire will tell the two scorers to figure it out. You can usually get together, go down the inning in question and figure it out reasonably. Almost always I've immediately had the other scorer ask something like, "Did Bill score in the 5th?" as they already have a question in their minds.

Only once have I not reached agreement and this was with another coach who claimed at the end of the game his team had 3 more runs than they'd earned. The umpire had a card and was keeping track and had declared a run rule. If they really had those 3 runs, it wouldn't have been. He wouldn't sign the card and I wouldn't accept 3 more runs against in pool play. He had the offical book. The umpire asked to look at each book, the other coach started coloring in really fast while the umpire looked at mine. Then the umpire looked at his and declared that his book didn't balance. He'd changed an out to a run then added an extra batter to the bottom of his order and had him hit a few HR's, all so he could lose 12-5 instead of 12-2 or something like that. I didn't get it. Umpire just shook his head and walked off. He reported my score.