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[Big Slick]

Originally Posted by DeputyUICHousto

I believe the BU has "intermediate" runners at 3B. I could be wrong.

No, as was discussed Here

Also, from the mechanics manual:
In the section marked "FP with runners on second and third" BASE UMPIRE.
. . .
4. On all balls hit through the infield or to the outfield that you do not go out on, immediately come inside the diamond, button hook and take the batter-runner all the way to third base.

And in the preamble to the two umpire system:
"The PLATE UMPIRE should be prepared to take a share of the base plays. The lead runner, if there is more than one on the bases, it he plate umpire's responsibility. . . . If there are two runners on base and the next batter hits for extra bases which will score the lead runner without a play, the plate umpire pays little attention other than noting that the runner touched third base and the plate. Attention then is directed to the second runner and the play that may be made. The BASE UMPIRE meanwhile ascertains that all runners touch second and First Bases (sic) and takes whatever plays are made on the batter-runner."

Furthermore:
"There are four times a base umpire will make a call at third base:
1. On the batter-runner on a triple with no runners on base.
2. One the last runner into third base.
3. On a lone runner on fly ball advancement.
4. On any return throw from the plate area or a cut-off by a player."

Therefore, in the OP, BU does not have responsibility on R2 once she advances to third, and must pick up the BR. If, in the OP, the catch was made, then BU would take R2 into third (that's a combination of #2 and #3).

Your comment may be referring to #4, but this is not the OP.