Tropical singularity persistence conjecture for smooth planar systems

Let a tropical dynamical system have a tropical singularity at $(u_0,v_0)$ of sink, source, or saddle type. Let $X$ be a neighborhood of $(u_0,v_0)$. Tropical singularity persistence conjecture. For all $0<\epsilon\ll 1$, the associated smooth system has a unique hyperbolic singularity at $(u_\epsilon,v_\epsilon)\in X$ such that $\lim_{\epsilon\to0}(u_\epsilon,v_\epsilon)=(u_0,v_0)$, and it has the same type: stable node, unstable node, or saddle. This conjecture asserts persistence of the local phase portrait under the smooth singular perturbation and is part of the paper's proposed connection between tropical and smooth systems; it is left unresolved.