Integral cohomology and non-isomorphism conjecture for generalized roofs

For $n=2k+1$, let $Y_1$ and $Y_2$ be the two smooth Calabi–Yau varieties arising as the zero loci in the generalized roof construction, each of dimension $k^2-1$.

Integral cohomology and non-isomorphism conjecture. One has an isomorphism of integral cohomology groups

$$H^{k^2-1}(Y_1,\mathbb{Z})\cong H^{k^2-1}(Y_2,\mathbb{Z}),$$

but, generically, the varieties are not isomorphic:

$$Y_1\not\cong Y_2.$$

The preceding theorem gives the corresponding rational cohomology isomorphism, while the integral statement and generic non-isomorphism are posed as a conjecture. The $k=2$ case is known to provide a derived-equivalent, non-birational example, whereas the general case remains unresolved in the supplied text.