I don't quite get that. As far as I understand it the "nothing at stake" problem works by a malicious party inducing a fork, one of which they double-spend in. Since it's in the best interest for everyone else to mine both forks, you can force your double-spend fork to become the longest chain by only validating the double-spend fork.

This means you have to trust that nobody part of your current chain has double-spent in this way. But isn't this the same as in PoW where you have to trust that nobody has launched a 51% attack to disrupt the network in the past?

Also, can't you just prevent people from mining all forks? I.e. for becoming a validator you have to deposit X as a security beforehand and you can only earn at most X via staking (so it is in the history before you can attack with nothing at stake). If it is recognised that you mine on more than one fork at a time, you lose the security deposit you gave before the fork. X goes to the person who found the fork, incentivising that the mallicious fork is identified on all forks (miners on competing forks are incentivised to look at all forks and quickly add the mallicious fork detection for their own benefit). If you want to retrieve your security and money earned, you have to announce this on all forks (you immediatly seize to be a validator). You are only allowed to retrieve the funds, if it is confirmed on all forks, or the forks are sufficiently behind the longest chain. This allows everybody ample time to look for dual-fork work and also incentivizes rapid solution of forks.