An HMAC depends on the Message, Key and the Hash function used such as SHA-256, SHA-512, etc.
Consider if the computation of the MAC used a different set of Initial hash values or initial array of round constants. Instead of the "first 32 bits of the fractional parts of the square roots of the first 8 primes 2..19", the first 8 primes 8675309 onward or similarly random non-trivial set of bits are used.
It appears that those values are not special, other than they should have little correlation. The usual specified values are well tested, unlike the discussed different sets.
A similar post addresses some impact, yet I'd like to go one step further.
Suppose these different Initial hash values (IV) and initial array of round constants are kept secret, otherwise the MAC is computed as usual.
What would be the negative authentication impact of exposing the secret cryptographic key of the HMAC in this case?
Is the key still needed?