Under the antitrust limitations in the Sherman Act Sections 1 and 2, please provide your comments in the scenario below:

Dillee, Inc. and Dallee, Co. sell nearly all of the pillows that are purchased in the United States. These companies have registered trademarks for "Dillee" and "Dallee," respectively.

Dillee and Dallee entered a contract under which each company agreed to not raise any trademark infringement claim against the other and both companies agreed to keep pillows at a certain price that is profitable for each.

Jared invents a new pillow-making machine, which he can use to profitably sell pillows at a cheaper price than Dillee's and Dallee's. Jared seeks a license from Dillee to use its trademark for his cheaper pillows. Dillee agrees, but only if he sells his pillows at Dillee's higher price. Jared refuses those terms and then makes the same proposal to Dallee, which also agrees to a license only if Jared sells his pillows at Dallee's higher price. Jared refuses.

Jared then places advertisements on large billboards at major intersections throughout Los Angeles, with the words "Jared's Pillows are just like Dillee's and Dallee's, but cheaper."

Dillee and Dallee sue Jared for trademark infringement. Jared raises antitrust violations as a defense. Dillee and Dallee argue that they merely exercised their right to refuse to grant a trademark license to a would-be competitor. Have Dillee and Dallee violated antitrust laws?