Ephemeral key is held constant in ElGamal Is there a known algorithm to recover the sender's messages in ElGamal $\mathbb{F}_p^*$ if the ephemeral key is held constant in two or more transmissions, assuming the messages are always distinct?

Say Alice picks a prime $p$ between $2^{1000}$ and $2^{1001}$ and chooses a $g \in \mathbb{F}_p^*$ with large prime order. $L$ is her public key. Doug the doofus sends Alice two or more transmissions $(c_1, c_i), c_1 = g^k (mod\,p), c_i = m_{i}L^k (mod\,p) i = 1,2,...,n$ with $k$ held constant. Can Eve the eavesdropper recover Doug's messages?

IOW, why does the ephemeral key need to be ephemeral?
One thing Eve can do is compute the modular division of any two messages.

That is, she is given both $c_2 = mL^k$ and $c'_2 = m'L^k$; she can then compute the value $c_2\cdot c'^{-1}_2 = m\cdot m'^{-1}$.

Whether this, in itself, is sufficient to recover the messages depends rather on the distribution of Doug's messages.  However, what this means (independent of the distribution) is that if Eve has a guess for the plaintext for any of the messages, she can immediately deduce the plaintext for all the other messages.

Note that this similar system IES doesn't share this problem; if Doug repeats all the parameters, well, it will leak if he resends identical plaintexts; however it will still be secure otherwise.