1.  Rollout^{1}  Bar/21 7/6  eq: 1.003 
 Player: Opponent:  29.14% (G:2.19% B:0.05%) 70.86% (G:35.26% B:1.06%)  Conf.: ± 0.004 (1.007...0.999)  [100.0%] Duration: 50 minutes 50 seconds 

2.  Rollout^{1}  Bar/21 6/5  eq: 1.026 (0.023) 
 Player: Opponent:  29.56% (G:2.10% B:0.04%) 70.44% (G:37.28% B:1.31%)  Conf.: ± 0.003 (1.029...1.022)  [0.0%] Duration: 43 minutes 29 seconds 

3.  Rollout^{1}  Bar/21 4/3  eq: 1.041 (0.038) 
 Player: Opponent:  28.21% (G:1.98% B:0.04%) 71.79% (G:38.11% B:1.33%)  Conf.: ± 0.002 (1.043...1.038)  [0.0%] Duration: 44 minutes 13 seconds 


^{1} 15552 Games rolled with Variance Reduction. Dice Seed: 39912497 Moves: 3ply, cube decisions: XG Roller

Mike Mannon gave an insightful analysis of the backgame position that he rolled out, shown above.
For the 4, bar/21 is forced. For the ace, Mike compared 7/6 to 6/5. For 6/5's side of the ledger, he stated, The net pickup is on 43, 42, and 41.
I think we should add to that the rolls of 32 and 22.
To me, the real key is in Mike's next paragraph: The opponent blots with 44, 55 and 64 on this turn...
This explains why 7/6 rolls out better than 6/5 by a clear margin (.022). If Blue might hit a shot on his next roll, there is more incentive to keep his board intact and sacrifice a bit in board conservation / flexibility.
In the first variant below, I've adjusted White's distribution so that no number forces her to leave a shot on the upcoming roll. In this case, Blue's 6/5, 7/6 and 4/3 are all "tied."
The second variant below (on the right, if your window is wide enough) is identical to the first except the cube has been moved to Blue's side. In this way, more games will be played out, and we can arguably get a truer sense of how the checker plays compare. Now 4/3 falls off sharply, and 6/5's tiny edge over 7/6 increases a little.
(In case anyone wonders, hitting with bar/24* 7/3 is nearly a doublewhopper.)
Nack
1.  Rollout^{1}  Bar/21 6/5  eq: 1.039 
 Player: Opponent:  28.30% (G:1.87% B:0.04%) 71.70% (G:36.83% B:1.37%)  Conf.: ± 0.002 (1.041...1.037)  [97.5%] Duration: 2 hours 04 minutes 

2.  Rollout^{1}  Bar/21 7/6  eq: 1.043 (0.004) 
 Player: Opponent:  27.44% (G:1.98% B:0.04%) 72.56% (G:35.87% B:1.25%)  Conf.: ± 0.003 (1.046...1.040)  [2.0%] Duration: 2 hours 04 minutes 

3.  Rollout^{1}  Bar/21 4/3  eq: 1.043 (0.004) 
 Player: Opponent:  26.98% (G:1.77% B:0.04%) 73.02% (G:37.85% B:1.41%)  Conf.: ± 0.002 (1.046...1.041)  [0.4%] Duration: 2 hours 41 minutes 


^{1} 15552 Games rolled with Variance Reduction. Dice Seed: 43835611 Moves: 3ply, cube decisions: XG Roller

1.  Rollout^{1}  Bar/21 6/5  eq: 0.670 
 Player: Opponent:  28.85% (G:1.81% B:0.05%) 71.15% (G:37.42% B:1.48%)  Conf.: ± 0.003 (0.672...0.667)  [100.0%] Duration: 2 hours 27 minutes 

2.  Rollout^{1}  Bar/21 7/6  eq: 0.677 (0.007) 
 Player: Opponent:  27.94% (G:1.95% B:0.05%) 72.06% (G:36.18% B:1.29%)  Conf.: ± 0.003 (0.679...0.674)  [0.0%] Duration: 2 hours 11 minutes 

3.  Rollout^{2}  Bar/21 4/3  eq: 0.721 (0.051) 
 Player: Opponent:  27.42% (G:1.68% B:0.04%) 72.58% (G:38.59% B:1.59%)  Conf.: ± 0.005 (0.726...0.716)  [0.0%] Duration: 47 minutes 22 seconds 


^{1} 15552 Games rolled with Variance Reduction. Dice Seed: 25831846 Moves: 3ply, cube decisions: XG Roller
^{2} 5184 Games rolled with Variance Reduction. Dice Seed: 25831846 Moves: 3ply, cube decisions: XG Roller
