Perfect Secrecy in Symmetric Key Systems For symmetric key systems does perfect secrecy imply that all ciphers in the cipher space are equiprobable?

I would think not, but I am finding it hard to come up with a counter example.
In general, no. Consider the following perfectly secret probabilistic encryption scheme. Encrypt exactly like a one-time pad. Then, with probability 1/4 add a 0-bit after the ciphertext, and with probability 3/4 add a 1-bit after the ciphertext. For decryption, first throw away the last bit, and then decrypt as in the one-time pad. The ciphertexts in the ciphertext space are now not equiprobable.

Having said this, Shannon's theorem says that when the message space, ciphertext space and key spaces are all the same size, then a scheme is perfectly secret if and only if every key is chosen with the same probability and every plaintext and ciphertext have a single key mapping one to the other. A simple corollary from this theorem says that in this case (where all spaces are the same size) then the ciphertext values are all equiprobable.