BIP-32 hardened children

[justusranvier]

Can an observer prove that a given public key is the n^th hardened child of a given xpub without access to any of the associated private keys if they know n?

Alternately, can the owner of the private keys create such a proof?

[johoe]

To prove that a key is the nth hardened child, you need to provide I_L and prove that it is the result of a HMAC_SHA512 operation.  I don't see how you can prove the latter without giving the second operand of this operation.  However, for a hardened step this contains the private key.