CHICAGO — A familiar sort of preservation battle has been stewing for months here over the fate of the old Prentice Women’s Hospital, a concrete, cloverleaf structure from 1975 by Bertrand Goldberg, the Chicago architect. It’s a groundbreaking, wonderful oddball among the architectural monuments in this city. High-profile designers like Frank Gehry, Jeanne Gang, Tod Williams and Billie Tsien have signed petitions entreating Northwestern University, which owns the building, not to tear it down, arguing for landmark status and pleading for Mayor Rahm Emanuel to step in.

The university says it needs new biomedical research facilities and that Prentice is too small, old and quirky to feasibly retrofit. A new building, the university argues, would bring to the city millions of investment dollars, create jobs and save lives.

No surprise, Northwestern has been winning the debate. On Monday Brendan Reilly, an alderman representing the Chicago ward that includes Prentice, announced that he was leaning toward demolition. “I remain open to suggestions,” he added, according to The Chicago Tribune. “And believe me, if there’s a eureka moment, I’m all ears.”

So here is a suggestion: Build a research tower on top of Prentice.

I brought up this notion with Ms. Gang, probably the most celebrated architect of the current generation here in Chicago, when we stopped to look at the building the other afternoon. I was curious about a strategy of accretion, layering. Although an advocate for preservation, Ms. Gang embraced the thought and ran with it. I’ll get to her plan in a moment.