The finite homological Lagrangian monodromy classification conjecture

Let $L$ be the monotone Lagrangian under consideration in dimension $n$, and let $\mathcal{H}_L$ denote its homological Lagrangian monodromy group. Assume that $\mathcal{H}_L$ is finite.

Finite monodromy classification conjecture. Either $\mathcal{H}_L$ is isomorphic to a subgroup of

$$\operatorname{GL}(n-1, \mathbb{Z}),$$

or there exist integers $n_1,\dots,n_k\geq 2$ satisfying

$$\sum (n_j-1)=n$$

such that $\mathcal{H}_L$ is isomorphic to a subgroup of

$$S_{n_1}\times\dots\times S_{n_k}.$$

This is proposed on the basis of computer experiments as a classification prediction for finite homological Lagrangian monodromy groups in dimensions greater than two. The preceding three-dimensional theorem gives the corresponding restriction to subgroups of $S_4$, $S_3\times S_2$, or $S_2\times S_2\times S_2$, but the general assertion remains open.