Fuglede's conjecture for elementary abelian groups

Let $p$ be a prime and $d\geq 1$. A set $E\subseteq (\mathbb Z/p\mathbb Z)^d$ is tiling if there exists $T\subseteq (\mathbb Z/p\mathbb Z)^d$ such that the translates $\{E+t\mid t\in T\}$ partition the group. It is spectral if there exists $A\subseteq (\mathbb Z/p\mathbb Z)^d$ such that the characters $\{\chi_{\mathbf a}\mid \mathbf a\in A\}$ form an orthogonal basis of $L^2(E)$, where $\chi_{\mathbf a}(\mathbf x)=e^{\frac{2\pi i}{p}\langle\mathbf a,\mathbf x\rangle}$. Fuglede's conjecture. A set $E\subseteq (\mathbb Z/p\mathbb Z)^d$ is a tiling set if and only if it is a spectral set. The conjecture is refuted in $(\mathbb Z/p\mathbb Z)^4$ for every odd prime by the paper's construction of a spectral, non-tiling set of size $2p$; the source also notes that it holds in $(\mathbb Z/2\mathbb Z)^4$.