Cost is not the issue, Each Block has a defined target of say 1 minute between blocks.
Your chain is 3 months behind, and still has a target time of 1 minute, your block height will always be ~ the same 3 months behind and as such never a threat to causing a reorg, because a reorg can only happen if your block height # exceeds the main chain.
So how do you make up the 3 months time difference?
FYI:
Any change to the code to modify the time target between blocks could allow faster blocks, would lower the target difficulty making it a weaker chain and also break consensus with the other nodes, therefore making sure it would never be accepted over the main chain.
FYI2:
The phrase (block production has zero cost) , is incorrect.
There actually is a cost , it is time.
Your block has to wait the coded time before block generation can occur, and those coins go dormant for a coded period, another time factor.
The Time between blocks is hard coded which affects the difficulty # in proof of stake coins, thus defining the strength or weakness of a chain.
What exactly would make a block of a POS coin invalid, e.g. timestamp too late, compared to timestamp of previous block?
A POW coin can have a target time of 1 minute but could be stalled for days. Some shitty ones regularly do this.
If a block has to wait the coded time of 1 minute before block generation can occur, then every node must have really exact system time. Not like Bitcoin
A timestamp is accepted as valid if it is greater than the median timestamp of previous 11 blocks, and less than the network-adjusted time + 2 hours. "Network-adjusted time" is the median of the timestamps returned by all nodes connected to you.
(quoted from wiki)