With PoS blockchain networks N@S can become problematic in case of a chain-split (forking).
For example, we get ourselves in chain-split over Segwit with Peercoin OG as the hostile splinter chain. Now, all the minters have equal stake on both of chains, and have no real incentives to “pick a winner” as minting on both chains is perfectly viable. Naturally, the selection of the “real Peercoin” will come down to exchanges and miners. Real Peercoin will become any chain which is adopted by exchanges and the chain on which miners remain active.
However in this scenario splinter chain can continue living indefinitely without any community support, users and real development. Which can be confusing to community, exchanges, miners and businesses using the chain. It can be dragged out in megabytes of forum discussions and twitter flame wars.
This is actually the most important aspect of Nothing-at-stake problem, inability to easily pick the real chain in case of the chain split.
Which made me think that if minters can claim the transaction fees, they will resort to support the chain with more transactions and in time their support for the lesser chain will diminish.