It depends whether you want to be able to stop and start and get the same answers as if you'd just kept going. If you're willing to change everything and restart with a new seed, sure.
I also figured at the interface level, the easiest thing to do would be to make them all available. At the very least, we need them for diagnostics.
How do you average covariance matrices to get the same result as if you'd thrown all the data together and estimated a covariance (assuming equal length chains)? I'm pretty sure you'd need the means (and counts?) too. We could do it at the level of sufficient statistics like the Welford accumulators.