A critical public hearing on a proposed hotel next-door to the Merchant’s House Museum has been postponed for the fourth time. A spokeswoman for the Landmarks Preservation Commission said that the presentation, originally scheduled for tomorrow, had been put off at the request of the developer, who needed more time to prepare. The museum’s administration has fiercely lobbied against the nine-story hotel, saying it would ruin the 180-year-old building’s aesthetic and could potentially undermine its foundation, as well. A new date for the hearing has not yet been scheduled. Update | 3:56 p.m. An earlier version of this post said the public hearing had been delayed for the second time. According to the Merchant’s House Museum, it is the fourth. “We want to believe it is because the Commission is taking this very, very seriously. As well they should,” the museum wrote in an e-mail.