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Do you want an All-Star Game? If seems like no since not wanting league-wide offdays. If yes, that could done by slotted in extra days at the middle break or one after.
But the scheme of starting the subleagues offset by a day... The problem there is that the interleague series then cannot line up exactly.
The interleague series can be positions adjacent to offdays so that they can be slid one way or the other. In those cases it becomes chunks of 4 and 2 series separated by an offday rather than always 3 and 3.
Or that can just be done in general with the subleagues starting at the same time. Then, e.g., the third series after which the first offday would fall - some of those could be slid forward a day to have the teams off before it rather than after.
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