  We study the cohomology of a 1-parameter family Y_t of Calabi-Yau 3-folds
introduced by Aspinwall and Morrison, related to the mirror quintic family.
Szendr\H{o}i proved that Y_t, Y_{xi t}, ..., Y_{xi^4 t}, where xi is a fifth
root of unity, have the same rational Hodge structure but are not isomorphic,
and conjectured that they are not birational or even derived equivalent. We
confirm this by proving that their integral Hodge structures are different, and
discuss how this fits with known Torelli-type theorems and counterexamples.
