No, this does not follow. Since the Hausdorff dimension dominates the Fourier dimension, it suffices to establish the existence of a compact set $K$ of Hausdorff dimension zero that supports a Rajchman measure.

The last theorem of section 3 (attributed to Ivashev-Musatov 1962) of Lyons's survery gives a considerably stronger version of this statement (note that $\mathcal U_0$ is defined in the introduction of the paper as the collection of sets that are annihilated by all Rajchman measures).

No, this does not follow. Since the Hausdorff dimension dominates the Fourier dimension, it suffices to establish the existence of a compact set $K$ of Hausdorff dimension zero that supports a Rajchman measure.

The last theorem of section 3 (attributed to Ivashev-Musatov 1962) of Lyons's survey gives a considerably stronger version of this statement (note that $\mathcal U_0$ is defined in the introduction of the paper as the collection of sets that are annihilated by all Rajchman measures).