Could a Proof of Burn function be implemented into Bitcoin Core to help prove loss of private keys? Is there a list of self reported BTC addresses with lost private keys? Of course such a list would never be conclusive as we would have to trust the people who report that the keys were lost.

Could this trust issue be eliminated going forward by creating a Bitcoin proof of burn function (forcing BTC to be sent to a random address where it is extremely unlikely nobody holds the private key). 

Such a function could not include the ability for the user to send BTC to an address of their choosing. It would need to be separate from the private key system already in place. It can also not be retroactive.

Assuming that the Private Key and "Proof of Burn Key" are stored separately, the purpose of this idea would be to allow:


Those who have lost their private key to prove their loss of ability to find it (this could have an impact on insurance, tax, inheritance claims or more)
You could use a multi-sig Proof of Burn system to help minority claimants (of blockchain businesses, heirs to a bitcoin estate, etc) the ability to call the bluff of a rogue partner or other beneficiary from making a "lost key" claim.
There's already a way to burn bitcoins. You simply assign the bitcoins to an OP_RETURN. No need for the complex machinery.