  We produce counterexamples to the birational Torelli theorem for Calabi-Yau
manifolds in arbitrarily high dimension: this is done by exhibiting a series of
non birational pairs of Calabi-Yau $(n^2-1)$-folds which, for $n \geq 2$ even,
admit an isometry between their middle cohomologies. These varieties also
satisfy an $\mathbb L$-equivalence relation in the Grothendieck ring of
varieties, i.e. the difference of their classes annihilates a power of the
class of the affine line. We state this last property for a broader class of
Calabi-Yau pairs, namely all those which are realized as pushforwards of a
general $(1,1)$-section on a homogeneous roof in the sense of Kanemitsu, along
its two extremal contractions.
