Question about proof of burn

You can institute any conversion rate you want.

That said, proof of burn is stupid. It ensures that people coming to your side chain lose an amount of value proportional to their issue of sidechain coins, but that value is lost - ie, it doesn't support the value of your sidechain coin at all, because you can't redeem it. People aren't "buying" the sidechain coins because the sidechain is gaining no value whatsoever from their sacrifice. If your sidechain crashes or ends or peters out, it's worth exactly the same as any altcoin that crashes or ends or peters out. IOW, with proof of burn only, it is just another altcoin, not a sidechain.

Now, if Bitcoin implements a way to show proof of burn in that other block chain and awarding bitcoins that have been "burnt" to your alternate block chain, then you have an actual factual side chain as opposed to an altcoin. But for it to be a side chain, coins have to be redeemable in both directions. And it's value will be absolutely pegged to the value of Bitcoin, less transaction fees for converting.