Topological determination by the log fiber for smooth proper exact morphisms

Let $P$) be a sharp toric monoid, let $V_P$ be a neighborhood of the vertex $v$ of the associated toric space, and let $f\colon X\to Y=V_P$ be a smooth proper and exact morphism of fine saturated log analytic spaces. For each $\rho\in V$ and $x\in X_v$, write $\lt{X_{x,\rho}}$ for the subgroup of the fiberwise torus acting on $\tau_X^{-1}(\tau_X(x))$, as defined by the quotient of the characteristic monoid by the face generated by the image of $F(\rho)$. Topological determination conjecture. After possibly shrinking $V$, there is a commutative diagram

\xymatrix{ X_{v,\mathrm{log}}\times V \ar[d]_{f_{v,\mathrm{log}}\times\operatorname{id}} \ar[r] & X_{\mathrm{top}} \ar[d]^{f_{\mathrm{top}}} \\ v_{\mathrm{log}} \times V \cong Y_{\mathrm{log}} \ar[r] & Y_{\mathrm{top}}, }

where the top arrow identifies $(x_1,\rho_1)$ and $(x_2,\rho_2)$ if and only if $\rho_1=\rho_2$, $\tau_X(x_1)=\tau_X(x_2)$, and $x_1,x_2$ lie in the same orbit under the action of $\lt{X_{\tau(x_i)}}(\rho)$ on $\tau_X^{-1}(\tau_X(x_i))$. In particular, the log fiber $f_v\colon X_v\to v$ determines $f$ topologically in a neighborhood of $v$. This gives a local topological description of a smooth proper exact log morphism in terms of its central log fiber; the source does not provide evidence that the claim has been resolved.