In https://crypto.stackexchange.com/questions/72969/proof-dlog-is-hard-in-generic-group-model/ it is shown if we allow only exponentiation and multiplication we can get an exponential complexity lower bound on discrete logarithm computation.

If in addition if we allow Diffie-Hellman operations as well, would there be any sort of exponential lower bound?

In https://crypto.stackexchange.com/questions/72969/proof-dlog-is-hard-in-generic-group-model/ it is shown if we allow only exponentiation and multiplication we can get an exponential complexity lower bound on discrete logarithm computation in the generic group model.

If in addition if we allow Diffie-Hellman operations as well, would there be any sort of exponential lower bound in the generic group model?