Run Scores on Double Play?

With one out and runners on first base (R1) and third base (R3), the hit-and-run is on, so the runners are moving on the pitch. The batter hits a line drive to the second baseman and it is caught for the second out. The second baseman then chases R1 back towards first base and is able to tag R1 before he returns safely to first base (this is the third out, inning over!).
 
R3 touched home before R1 was retired over at first base, but R3 clearly never properly tagged up at third base. Was R1's third out over near first base a force out? If so, R3's run would not count. 
 
Since R3 did not properly tag-up, should his run count?
 
Call
A force play is a play when a runner loses his right to the base he occupies and is forced to advance when the batter becomes a batter runner (NF 2-29-3).
 
If a runner leaves base too early, he is subject to being declared out upon proper and successful appeal (NF 8-2-5). An appeal may be made after the third out, as long as it is made properly (NF 8-2-6i).
 
In the play below, the batter was retired on his line drive for the second out of the inning. R1 was not forced out for the third out of the inning, so any runs that score during the play would count (unless properly appealed).