Evgeny,
I change my server http channels from synapse to Indy a few months ago, to allow the server compile on delphi/linux64. Work fine on linx, the only complain is have some memory leaks i cant detect yet.
Form that change i detect the connection down after a time without activity at the client app. I have activated disablenagle and keepalive on the server. Theres any another setting i must check?