Tiles with no spectra in dimension 4

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Multiple lattice tiles and Riesz bases of exponentials (2013, background): “The so called Fuglede or Spectral Set Conjecture [3] (claiming that for Ω to have such a basis it is necessary and sufficient that it can tile space by translations) was eventually proved to be false in dimension at least 3 [15, 8, 1, 2], in both directions.”
Weak convergence and spectrality of infinite convolutions (2022, background): “Although the both directions of Fuglede’s conjecture were disproved by Tao [52] and the others [22, 23, 32, 33, 43] for d ≥ 3, the connection between spectrality and tiling has generated a lot of interest, and some affirmative results have been proved for special cases [29,34].”
Construction, classification and parametrization of complex Hadamard matrices (2011, methodology): “We refer the interested reader to the survey article [86] and to the recent literature, including the papers [50], [69] and [94].”
On Fuglede's Conjecture and the Existence of Universal Spectra (2006, background): “A bounded open set T ⊆ G is called spectral if it possesses a spectrum.”