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I think this will be available every week, each week a chance to give a gift to the Brass Embassy.
The goal is to raise your 'A Shaper of Starved Culture' to 6, so you can receive the token from the Starved Men.
I believe it's a similar concept to Mrs Chapman's Boarding House: you could only explore the house once per week. Until your Acquaintance with her has maxed out.