For now, we should point out that proof-of-stake mining has inherent problems, which, to my knowledge, neither BCNext nor anyone else in the Nxt community has addressed. In particular, the well-known "nothing at stake" attack, by which miners can mine on multiple blockchains at no cost to themselves; this is in contrast to proof-of-work, where the resources one uses to mine are depleted in the process of mining. "Transparent forging" doesn't solve this problem, and, in any case, transparent forging has not even been implemented yet.
This is what a Nxt dev had to say about the "nothing-at-stake" problem:
If you have a computer that can forge a very large number of accounts in parallel without using any CPU and can also fund these large number of accounts without costing any NXT, then sure nothing is at stake
In my experience though you need to fund each separate acct with NXT (wait long enough) and then even a small amount of CPU time, multiplied by millions of accounts, well that seems to take some CPU power after all.
So, if you ignore the millions in NXT fees to create the accounts and the CPU power needed to forge on those millions of accounts, then you can try to monkey around with grabbing all the forging fees.
economically it doesnt make sense as there is NXT at stake. I think the premise of "nothing at stake" is that it rhymes with "proof of stake" so it has to make sense. It seems the assumption is that they can buy an arbitrary amount of NXT for free, so maybe much closer to genesis this would have been an issue, but now it is nonsensical.
That has nothing to do with mining on multiple chains at once.