Right, so if you had a fixed dice roll in the past and translated that into the measurement results on each axis in a static way, that wouldn't work. You have to make a fresh random dice roll after the experimenter chooses which axis to measure - or you have to translate the past dice role into the result for the axis in a way that depends on which other axes the experimenter chose to measure.

I assert that this is not terribly surprising, and Conway is actually just doing a sleight of hand around the definition of "random". We would normally expect a truly random event to be (by definition) uncorrelated with anything else, in this case including counterfactual versions of itself - the random measurement you get from a given axis must not be correlated with the measurement you would have got if you'd measured a different combination of axes. That's maybe a little odd, but I don't think it contradicts people's normal notion of "randomness", particularly in a QM context. It's like how in early online poker games people would cheat by figuring out the "random seed" and know all the cards - because that's not real randomness.