Existence of sets with finite extended Fourier spectrum strictly above the ambient dimension

Determine whether there exists a compact set X ⊆ R^d and a parameter θ ∈ [0,1] such that the extended Fourier spectrum E_F^θ(X), defined as the supremum over all finite Borel measures μ supported on X of the Fourier spectrum dim_F^θ(μ), satisfies d < E_F^θ(X) < ∞ (equivalently, d < E_F^θ(X) < 2d as permitted by general bounds).

Background

The paper introduces the extended Fourier spectrum E_F^θ(X) of a set X ⊆ R^d as the supremum of the Fourier spectra dim_F^θ(μ) over all finite Borel measures μ supported on X. For sets of Lebesgue measure zero, E_F^θ(X) coincides with the usual Fourier spectrum, while for sets with non-empty interior E_F^θ(X)=∞ for all θ.

General bounds imply E_F^θ(X), E_F^θ(X) ∈ [0,2d] ∪ {∞}. The authors point out that they do not know any example where E_F^θ(X) is strictly larger than d but still finite, prompting the existence question below.