The Starks are yet to declare for Blacks in the Dance of the Dragons, but House of the Dragon - Season 1 clearly states that they are not oathbreakers. Queen Rhaenyra Targaryen wishes to know her allies before declaring war on the Greens and to that end, sends her heir, Prince Jacaerys Velaryon, to the Vale of Arryn and Winterfell. She also sends her secondborn Prince Lucerys to the Baratheon seat of Storm's End.