One security issue of original convergent encryption is, it is vulnerable to off-line brute force attack. If the adversary knows the entire message space, it can sample each message, computes the hash, encrypts with the computed hash (key) and compares the ciphertext of sampled message with the target ciphertext. If both are the same, the adversary can deduce the sampled message equals the message underlying the target ciphertext. This type of attack has been recognized by Bellare et al., and they formalised a semantic security definition under unpredictability assumption (i.e., not allow the adversary to predict and sample message). The following work DupLESS prevents the off-line brute force attack by introducing a third party entity for co-generating the encryption key, i.e., the encryption key depends on both the message content and a system-wide secret key, which is kept by the third party, so that the outside adversary (not accessing the third-party entity) cannot launch off-line brute force attack.
Another (possible) weakness of convergent encryption is, it might be vulnerable to statistical attack. Although the RCE scheme (having been summarized in one answer before) can generate random ciphertext, the tag for message must be deterministic. The tag consistency defined in Bellare et al. requires, if two messages are the same, they must have identical tag. The tags essentially reflect the probabilistic distribution of plain messages. If the message space is with limited min-entropy, and the adversary has some pre-knowledge about the distribution of the message space, it might successfully guess some messages with significant probability.