Kitaoka's spectral commensurability conjecture for flat tori

Let $L_1$ and $L_2$ be lattices in Euclidean $n$-space $\mathbb{R}^n$. The corresponding flat tori are $\mathbb{R}^n/L_1$ and $\mathbb{R}^n/L_2$; they are spectrally commensurable when, up to rescaling of the metrics, their spectra are mutually contained in each other. Two lattices are commensurable if their intersection has finite index in each.

Kitaoka's conjecture. If the flat tori $\mathbb{R}^n/L_1$ and $\mathbb{R}^n/L_2$ are spectrally commensurable, then, up to an isometry of $\mathbb{R}^n$, the lattices $L_1$ and $L_2$ are commensurable.

This conjecture refines a question relating the spectrum of flat tori to their arithmetic, viewed through isogeny and commensurability. No resolution is supplied in the source.