I finally got an answer from support. They said that my eth deposits were cancelled due to "insufficient gas" and that my coins should be back at the sending address. They are not, but now I'm not sure that the problem was Poloniex fault, it could be HitBTC's fault (I sent the ether from there) or it can be a bug in the Ethereum protocol.

In any case, taking three weeks to look into a customer's request for help is unacceptable.

I have already withdrawn all my money from Poloniex, except for that Ethereum deposit that they have spirited away. As soon as I can get hold of that, I will withdraw it and never again trust them with my money. Until then, I will try to nag them on social media as much as I can.

Really? I only know @poloniex that is not very active, but it is true that many people try to contact them using that Twitter account, just because their support is useless and there's no other way of contacting them. There is also @MickD (CXO at Poloniex)

I have 2FA enabled, but they are still holding an Eth deposit I made two weeks ago, with thousands of confirmations. No reply from support either. I can't see any reason for this other than they want to steal funds from customers. I swear never to use Poloniex again.

I found some names of managers at Poloniex:
Curtis Hale, general manager
Tristan D'Agosta, Founder
Anthony Eufemio, Founder
Mick D @MickD CXO