Crepant resolution theorem for the heterotic/F-theory fiber product

Let $W_{4}'$ and $B_{3}'$ be the indicated models, let $\tilde{\tilde{\theta}}$ be the rational map in the commutative diagram, and consider the fiber product $W_{4}'\times_{B_{3}'}\mathfrak{h}_{SU(5)_{gauge}}^{\mathbb{C}}$. Crepant resolution theorem. The closure of the graph of the rational map $\tilde{\vartheta}$ in the diagram is a crepant resolution of

$$W_{4}'\times_{B_{3}'}\mathfrak{h}_{SU(5)_{gauge}}^{\mathbb{C}}.$$

The source marks this assertion as resolved; its proof traces the exceptional fibers of a crepant resolution back to the corresponding $E_{8}$ roots using the Brieskorn–Grothendieck equivariant resolution.