Gross–Ruan topological conjecture for special Lagrangian torus fibrations

Let $\pi: Z\to B$ be a special Lagrangian $T^3$ fibration of a compact Calabi--Yau threefold with respect to a Calabi--Yau metric whose compatible complex structure is sufficiently close to a boundary point with maximally unipotent monodromy, and whose Kähler class is sufficiently deep in the Kähler cone. The discriminant locus retracts onto a trivalent graph $D$, the monodromy around each edge is conjugate to

$$M=\begin{pmatrix}1&0&1\\0&1&0\\0&0&1\end{pmatrix},$$

and the vertices have positive and negative types with the stated fixed-plane intersections and monodromy matrices. Moreover, there are neighborhoods $U_P\subset B$ of the vertices such that fibers outside these neighborhoods have Euler characteristic $0$, while $\chi(\pi^{-1}(U_P))$ is $1$ or $-1$ according as $P$ is positive or negative.

Gross–Ruan topological conjecture. The discriminant and monodromy data of the fibration satisfy properties (i)--(iv) above, including the specified positive and negative vertex models.

This conjecture describes the expected combinatorial structure of special Lagrangian torus fibrations relevant to mirror symmetry. It is attributed in the source to Gross and Ruan; the supplied text gives no resolution evidence.