Adiabatic limit conjecture for co-associative Kovalev–Lefschetz fibrations

Fix a cohomology class $[\phi_{0}]\in H^{3}(M)$ and set $\chi=b([\phi_{0}])$. Let ${\cal H}_{\chi}$ be the corresponding flat affine bundle over the base $B$ of a co-associative Kovalev–Lefschetz fibration $\pi:M\rightarrow B$. A section is said to avoid excess $-2$ classes if its restriction to $B\setminus L$ avoids $-2$ classes and, at each point of $L$, the only elements of the relevant set ${\cal C}$ orthogonal to the vectors $v_{0},v_{1},v_{2}$ are $\pm\delta_{\alpha}$. Write $[B]\in H^{3}(X)$ for the pull-back of the fundamental class of $B$.

Adiabatic limit conjecture. If there is a positive section $h$ of ${\cal H}_{\chi}$ which avoids excess $-2$ classes, then for sufficiently large $R$ there is a closed positive $3$-form on $M$ in the class $[\phi_{0}]+R[B]$ with respect to which $\pi:M\rightarrow B$ has co-associative fibres. If, in addition, the section $h$ can be chosen to be maximal, then for sufficiently large $R$ the positive $3$-form can be chosen to define a torsion-free $G_{2}$-structure.

This is the proposed adiabatic limit problem for constructing co-associative fibrations and torsion-free $G_{2}$-structures from suitable positive sections. The source does not state a resolution, so the conjecture is recorded as open.