Force out or not; score the run?

[KenThree]

Need help on a play.

Runners on 2nd and 3rd with one out.

Batter hits fly ball that is caught by center fielder.

Runner on 3rd tags up properly, while runner on second runs like there are two outs, not tagging up.

Center fielder throws to second, but throw is not caught and rolls into infield. Pitcher picks up the ball and throws to second baseman before runner can return to 2nd base.

While this happens, runner on 3rd scores.

Is the out a 2nd a force out and the run does not count? Or is it not a force out and the run counts?

As you can see from my log-in, I am a long-time member of the forum as a basketball official and not a fan boy. So I knew that coming here would get me the right answer. That's all I need.

Thanks