LTP-strong conjecture for crepant Weierstrass fibrations

Let $B$ be a smooth compact manifold, let $W$ be a Weierstrass fibration in

$$Z=\mathbb{P}(\mathscr{O}_B\oplus\omega_B^{-2}\oplus\omega_B^{-3}),$$

and let $\widetilde{W}\to W$ be a crepant resolution obtained by a sequence of blowups of $Z$ in smooth centers, followed by taking the proper transform of $W$. Write $\widetilde{Z}$ for the resulting blown-up ambient space, and let

$$i:\widetilde{W}\hookrightarrow\widetilde{Z}$$

be the inclusion. LTP-strong conjecture. The inclusion induces an isomorphism of rational Hodge structures

$$i^*:H^k(\widetilde{Z},\mathbb{Q})\longrightarrow H^k(\widetilde{W},\mathbb{Q})\quad\text{for}\quad k<\dim W.$$

This is the stronger form of the numerical LTP-weak conjecture and is formulated after an example showing that an integral Hodge-structure isomorphism can fail because of torsion. Its status is not resolved in the supplied text.