The ambient-monodromy realization conjecture for toric fibres

Let $X$ be a compact toric manifold, let $T(x),T(x')\subset X$ be toric fibres, and identify $H_2(X,T(x))$ with $H_2(X,T(x'))$ canonically. Let $\mathcal{D}(x)$ and $\mathcal{D}(x')$ denote the distinguished subspaces of relative homology classes, and let $m_{T(x)}$, $m_{T(x')}$ and $\sigma_{T(x)}$, $\sigma_{T(x')}$ be the Maslov and symplectic area classes. An ambient monodromy is the map on relative homology induced by a Hamiltonian diffeomorphism mapping $T(x)$ to $T(x')$.

Ambient-monodromy realization conjecture. An isomorphism $\Phi\in\operatorname{Aut}H_2(X,T(x))$ can be realized as the ambient monodromy of a Hamiltonian diffeomorphism mapping $T(x)$ to $T(x')$ if and only if

$$\Phi(\mathcal{D}(x))=\mathcal{D}(x'),\qquad \Phi^*m_{T(x')}=m_{T(x)},\qquad \Phi^*\sigma_{T(x')}=\sigma_{T(x)},\qquad \Phi\vert_{H_2(X)}=\operatorname{id}.$$

The displayed identities are proved in the paper to be necessary for ambient Hamiltonian monodromy. The conjecture asserts that they are also sufficient; its general status is open.