The barns were demolished, and Mr. Davis planned to gut-renovate the Greek Revival, turning it into a guesthouse to accompany the 8-bedroom 11-bath main house, which was to have an elevator, an outdoor pool, a pool house and a tennis court.

But the Foursquare still needed a home. First he tried giving it away. When that didn’t work, he got permission to demolish it. Then the Peconic Land Trust stepped in, offering to trundle the 100-ton structure not far away to a 10.7-acre patch of unprotected land donated in the 1970s by Ronald Lauder, a son of the cosmetics tycoon Estée Lauder.

The land trust’s president, John v. H. Halsey, saw the benefits of this move as threefold: the house would be saved for its “agricultural history and rural character”; its placement would save the balance of the 10.7 acres from development; and the trust could use proceeds from its eventual sale to support a nearby farm.

Mr. Davis helped defray the house’s $55,000 moving costs.

Since then, events have not unfolded in quite the way anyone might have anticipated. To start with, the Foursquare had to be moved a second time, to another corner of the 10.7-acre plot, after a neighbor objected to its proximity, suing to have it at least 300 feet from her property.

In addition, enough residents questioned not only the house’s historic value but also the logic of where it was placed, that a civic group, Concerned Citizens of Sagaponack, was formed in opposition to its presence. Paul Brennan, a past president of Southampton’s Community Preservation Fund and a founding member of the new group, describes the Foursquare style as “dime a dozen,” and disputes the idea that the land it was moved to was in any danger of development, calling it “gifted open space” that had “been that way for 30 years.”

Another action his group questions is the land trust’s intention to put the house and its acre of land on the market, which they have been since last month, listed by Brown Harris Stevens for $2.5 million — even as the house still sits on the trailer used to convey it to the site. A foundation needs to be dug; steps and porches are missing; the electrical, wiring and plumbing must be redone.

Emma Clurman, the listing agent, said an application to expand the house 1,000 feet, most likely to the rear, could be tailored to accommodate a two-car garage and a pool. “The point is to maintain the classic architectural character of the house,” meaning the front, she said.

But Mr. Brennan views the idea of enlarging the house as compromising its historic value — and as proof that its significance was overblown in the first place, in order to profit from an eventual resale. “It’s a Foursquare on steroids, leaving us with an edited piece,” he said. “It was saved to sell.”

Mr. Halsey, for his part, said the resale of the house would help stem the tide of nonfarmers buying farmland, just “because they can,” and turning it into lawns, surrounding it with hedges or putting it to equestrian rather than agricultural use. About $700,000 of the proceeds from the resale, he explained, would be used to help a local farmer buy and protect Pike’s Farm nearby.

“A farmhouse next to a farm field feels right to me,” Mr. Halsey said, “especially if we are able to get more land into the hands of farmers. We are losing an acre here, but it is going to enable us to protect more acres.”

Foursquare homes, also known as Prairie Boxes, have central dormers and pyramid-sloped roofs. They were built all over the United States from 1895 through the 1930s. This one has four bedrooms and one and a half baths.

Ann Sandford, the chairwoman of the village’s architectural and historic review board, said there were a handful of Foursquare farmhouses left in the immediate area. The board originally found this one “worth saving,” because it “had architectural integrity,” with five sloped rooflines and harmonizing slopes on the porches, as well as an entryway, and had been “very solidly built.”

If an application is filed to expand the house, Ms. Sandford said, the board will “make a decision relative to what kind of additions a homeowner would want.”

Meanwhile, the village’s planning board has set an informal meeting on the house for Monday at 3 p.m. The land trust is encouraging residents to attend.

And in a gesture that he said was meant to “defuse the animosity,” Mr. Davis said he recently made Mr. Halsey an offer to buy back the farmhouse for its full asking price, move it to a “more compatible” half-acre lot that he owns nearby, renovate and resell it, and then build another house on the farmhouse lot.

Mr. Halsey said he thanked Mr. Davis for his offer but turned him down. “The whole concept here has been to give up an acre to protect a farmhouse to protect more land,” not to maximize profits, he said.