The dual spectral-set conjecture for spectra and tiling sets

Let $L\subset\mathbb{R}^{d}$. A spectrum is a set $L$ for which there exists a set $\Omega$ such that $(\Omega,L)$ is a spectral pair, and a tiling set is a set $L$ for which there exists a set $\Omega'$ such that $(\Omega',L)$ is a tiling pair. The dual spectral-set conjecture. $L$ is a spectrum if and only if $L$ is a tiling set, i.e., there exists a set $\Omega$ so that $(\Omega,L)$ is a spectral pair if and only if there exists a set $\Omega'$ so that $(\Omega',L)$ is a tiling pair. This is formulated as the dual counterpart of Fuglede's spectral-set conjecture; its general validity remains open.