To score or not to score

Question
Bases loaded with no outs.  Batter hits a ball to the outfield.  The runner on third tags up, but the runners on first and second do not.  The catch is made in the outfield for the first out.  Outfielder throws to first who tags the base for the second out.  First basemen throws home in an attempt to put out the runner from third.  Catcher makes a heads up play and without trying to tag the runner coming home throws to second to put out the second base runner who has not tagged up.  Short Stop touches second base for the third out, but the runner from third has already crossed the plate.  Does the run score???

Answer
This is a commonly misunderstood rule.  Each time you get a runner out for not tagging up in this situation, it is considered a live ball appeal.  The rule book states it as the following:  If a runner leaves a base too soon on a caught fly ball and returns in an attempt to retag, this is considered a time play and not a force out.  If the appeal is the third out, all runs scored by runners in advance of the appealed runner and scored ahead of the legal appeal would count.  So as long as the runner scores before the appeal out is made on the last runner, his run would count.  I hope this answers your question.  Feel free to leave feedback :)