Good point - I'd forgotten that the primaries are all run by the states, not by parties like the caucuses. So it doesn't matter if it's on one day or not, they

Good point - I'd forgotten that the primaries are all run by the states, not by parties like the caucuses. So it doesn't matter if it's on one day or not, they should be able to have a master list of "has voted" that gets marked no matter which party ballot you submitted. So, the only scenario where you need to know which primary they voted in is for states with an open primary that can also have a run-off election, which would give you the option of voting in a D primary first round and the R primary runoff if they didn't keep track (e.g Georgia has this system - https://www.wabe.org/party-matters-in-georgias-primary-runoff-election https://www.wabe.org/party-matters-in-georgias-primary-runof...)