Was it really mere coincidence, the Pozzuolis asked, that Mr. Zarco had availed himself of the exact same design — the one with “Joe’s” in a red, slanting script and “PIZZA” in block text set against a solid wash of white? Mr. Pozzuoli Jr. sent Mr. Zarco a cease-and-desist letter when he learned about the sign. Mr. Zarco changed it, but only slightly — straightening the “Joe’s.” The lawsuit quickly followed.

In January, Judge Cogan issued an opinion enjoining JPV from using the duplicative design, saying that Mr. Zarco’s version of events struck him as “dubious.” Judge Cogan found that the sign was an attempt “to create the impression among customers and potential customers” that the two pizzerias were “associated.”

In his opinion, the judge said that more than the logo had been stolen. Mr. Zarco, he explained, had adorned the walls of JPV with photographs of Mr. Maguire and others that had actually been taken at the Manhattan pizzeria; he had also posted images on JPV’s social-media accounts of “Spider-Man 2” — Mr. Maguire’s hit movie — being filmed at JP.

You might have thought that all this evidence, in a federal judge’s order, would have ended the fight. But as in other conflicts, a cease-fire in a pizza war isn’t easily achieved. Within three weeks of the order being issued, the Pozzuolis had already accused Mr. Zarco of ignoring it.

In a letter to Judge Cogan, the Pozzuolis claimed that while their rival’s sign had indeed changed — barely — his Twitter page was still using their logo, which had also appeared on Mr. Zarco’s GrubHub account. They further said that JPV had borrowed the one-of-a-kind “soccer dog logo” they had briefly used to promote their pies during the World Cup. As for the photograph of Mr. Maguire, it had not been taken down, they claimed. In fact, it was “enlarged.”