TL;DR Currently bitcoin hash rate is tied to worth because you're rewarded in a fashion directly related to their worth. In the future hash rate is tied to transaction frequency, which is independent of the network worth and hence need for security.
Thoughts?
Currently, about a million bitcoins per year goes for security. After 2140, and in practice much sooner, if you assume 1% transaction fees, to get the same level, you need 100 million in volume.
This means the security will probably go down. Parked bitcoins become freeloaders on the network. The network lives or dies depending on the transaction volume. Miners stop accepting free transactions.
There is also added complexity of bitcoin price. If most of the bitcoins is parked, value of those bitcoins (as well as all others) is determined by a small part that transacted.
Transaction fees will always push the bitcoin exchange price down, and if there are no other factors, bitcoin price will tend to zero. By "no other factors" in this case I mean no speculative buying and selling, and no round trip transactions. This is a silly case, because there is nobody buying.
If we add round trip transactions, the stable price is where the value added trough both kind of transactions is equal to the mining fees, and thus indirectly, security. More value added, more security.
If we magically again remove round trip transactions and add speculators and savers, the stable state is where fees equals new speculative amount. Bitcoins keep value as long as all saver keep buying a percentage of their value each year. If they don't do it, bitcoin price again goes down.
Another problem with bitcoin as a store of value where the security is determined by a minority that pays transaction fees, is that the liquidity is provided by the round tripping part, and they are disconnected from the price. So it means that hoarders save on the fees but do get hit by the volatility they themselves are creating.