when you stop converting into the main chain and you have lots of other more valuable chains this is called inflation.
I'm still unsure where miners get there intensive to support a chain that docent provide a mining investment. when you can explain that you'll have me.
you'll have to phrase the first one differently because I don't quite get what you're saying. what do you mean by "stop converting into the main chain". how do you qualify more valuable chains? more txs?
As for your question.
Bitcoin has one advantage over sidechains - it has the subsidy, and full node security, so I'm sure it'll be able to defend itself against abandonment or insecurity, and sidechains depend on bitcoin anyway so all bitcoin users on which ever chains have a meta-incentive to see bitcoin main remain secure. We have decades of subsidy ahead to deal with fee-only security for bitcoin, and sidechains may move forward the ways to do that because the sidechain by default has only fee security from the start.
You expect that all txs converge to one chain and create THE most valuable chain for miners to mine but in a world of sidechain how does that work? Is it the anonymous chain? Is it the fast tx chain? How does the miner decide? Maybe... just maybe... he should mine... ALL of them. Does that not make sense to you?