I do not know what is possible in a room, especially a small room, though it is possible in electronics. There are often equivalents between electronic and mechanical systems, though it seems to require less brain sweat to engineer an electronic system than a mechanical/acoustic equivalent.

A proper allpass filter ought to be flat, but allpass filters can have very long decay time or ringing.

For instance one can make electronic reverb out of allpass filters, and make them ring until next tuesday if you want.