Range proof for elgamal ciphertext

Alice has an ElGamal public key $y=g^x$. Bob encrypts a value $g^b$ based on Alice's Elgamal public key and he ends up with a ciphertext $(g^by^r, g^r)$. Can Bob prove that the value $b$ is in some range without revealing it or do you need to be the "owner" of the ElGamal secret key $x$ to create such proofs?

If $g^b$ is confusing then ignore it and consider a value $b$, I just need to know if I can create a range proof without knowing the $x$.

If your method of mapping your value $b$ to a group element is $g^b$, then creating a range proof for an El Gamal encryption is exactly the same as creating a range proof for a Pedersen commitment.

With El Gamal, you have $g^by^r$ where $b$ is the value, $r$ is the sender's ephemeral private key, and $y$ is the recipient public key.

Interpreted as a Pedersen commitment, you have $g^by^r$ where $b$ is the value, $r$ is the blinding factor, and $y$ is the alternative base point for which the discrete log w.r.t. $g$ (i.e. $x$) is unknowable to the committer/sender.

Note that since the recipient knows $x$, they can forge range proofs.

Details of how to create a simple range proof are here.