The orientation conjecture for stable-pair moduli on Calabi-Yau fourfolds

Let $f:\mathcal{X}\to\mathcal{B}$ be a smooth projective morphism with connected fibres of relative dimension $4$ to a smooth connected affine scheme $\mathcal{B}$, with $\omega_{\mathcal{X}/\mathcal{B}}\cong\mathcal{O}_{\mathcal{X}}$. Let $\widetilde{v}\in\bigoplus_p F^pH^{2p}_{DR}(\mathcal{X}/\mathcal{B})$ be a horizontal section. Orientation conjecture. There exists a finite étale cover $u:\mathcal{B}'\to\mathcal{B}$ such that the symmetric complex $\mathbb{E}$ restricted to ${\curly P}^{(q)}_{\widetilde{v}'}(\mathcal{X}'/\mathcal{B}')$ is orientable, where $\mathcal{X}'=\mathcal{X}\times_{\mathcal{B}}\mathcal{B}'$ and $\widetilde{v}'=u^*(\widetilde{v})$; moreover, the degree of $u$ can be chosen to divide $\lvert K^0_{\mathrm{top}}(\mathcal{X}_b^{\mathrm{an}})_{\mathrm{tor}}\rvert!$. This relative orientation is needed for deformation invariance of numerical invariants, while a canonical orientation is available after passing to a twofold étale cover in the relevant auxiliary construction.