The monodromy-compatibility conjecture for phase tropical hypersurfaces

Let $H_f$ be the complex hypersurface associated with $f$, let $\mathcal T\mathcal H_{\eta,f}$ be the corresponding phase tropical hypersurface, let $\psi:H_f\to\mathcal T\mathcal H_{\eta,f}$ be the map furnished by a parametrized version of the main theorem, and let $\xi_\eta:H_f\to H_f$ and $\mathcal T\xi_\eta:\mathcal T\mathcal H_{\eta,f}\to\mathcal T\mathcal H_{\eta,f}$ be the respective monodromy automorphisms.

Monodromy-compatibility conjecture. The diagram with horizontal maps $\psi$, vertical maps $\xi_\eta$ and $\mathcal T\xi_\eta$, and the corresponding map $\psi$ on the bottom commutes up to isotopy:

$$\mathcal T\xi_\eta\circ\psi\simeq\psi\circ\xi_\eta.$$

This conjecture asserts that the explicit automorphism of the phase tropical hypersurface models the monodromy of the degeneration under the comparison map. The source does not state that it has been proved or disproved, so the database status is open.