The new Conrad Athens, The Ilisian, is housed in the former Hilton Athens, which defined the Ilisia neighborhood for decades.Conrad Athens, The Ilisian.Credit: Conrad Athens The IlisianConrad Athens, The Ilisian Design throughout celebrates the hotel’s status as a modernist landmark, from the circular bar in the Athenian Lounge to the kidney-shaped end tables in guest rooms.A private running track on the mezzanine rooftop, inspired by the High L…