How to prove lifted ElGamal encryption with square root communication?

I'm looking for a more efficient solution to prove the correctness of multiple ciphertexts sent to different parties are correct.

The background is that $P_i$ uses lifted ElGamal encryption to encrypt a message $x_j$ to party $P_j$, where $j\in[N]$. Therefore, the ciphertext will be $Enc_{pk_j}(x_j;r_j)$.

Now I need to generate a proof for $P_i$ to show all the ciphertexts he generated is correct.

Since the second part of encryption will return me a value like $g^{s_j}\cdot (PK_j)^{r_j}$, I can't use batch schnoor identical scheme to prove the knowledge of all $s_j$ and $r_j$.

I think Gneralised Pedersen Commitment can be used here. for example, in the first move, receiver will return $\sqrt{N}$ commitment key, then prover generate $\sqrt{N}$ commitment.

But I'm still confused about the construction,