I'm not an attorney, but if I understand correctly, if one party gets to officially supplement the record, the other party gets a shot at rebuttal.
Aldisert (I believe it was) gave the State 10 days to file. Gura will have the same time frame (in other words, they will both file on the same day). Generally, the party making the supplemental record will share that record with opposing counsel a day or two before they file. Should the party asked to supplement the record file early, then opposing counsel will just have to pay attention (ECF will notify the opposing party via email) and get the rebuttal brief in on time. This kind of courtesy is a given.
We've seen this in a couple of other cases, albeit at district level.