Product spectrality conjecture

Let $A \subset \mathbb{R}^n$ and $B \subset \mathbb{R}^m$ be bounded, measurable sets, and let $\Omega=A\times B$. Product spectrality conjecture. The product $\Omega$ is spectral if and only if both $A$ and $B$ are spectral sets. The forward implication was posed as a question in earlier work and is known when one direction is obtained from spectral factors; the converse in full generality remains open.