Spectrality of tiles in powers of p-groups

Let $p$ be a prime, and let $n,d\geq 1$. Consider the finite abelian group $\mathbb{Z}_{p^n}^d$. The p-group tiling conjecture. Every tile in $\mathbb{Z}_{p^n}^d$ is a spectral set. This conjecture is proposed because the existence of non-spectral tiles in groups $\mathbb{Z}_{p^n}^d$ is not known. The paper proves Fuglede's spectral set conjecture for $\mathbb{Z}_{p^2}\times\mathbb{Z}_p$, but the stated assertion for all $n,d\geq 1$ remains open.