Well-poised hypersurfaces

Consider a monomial-free ideal $I$ in the polynomial ring in $n$ variables over an algebraically closed field. Following the paper [1] for every point from $\mathbb{R}^n$ I will define the initial form of the ideal $I$. The zero locus of the initial ideal is a flat degeneration of affine variety $V(I)$. The tropical variety $\mathrm{Trop}(I) \subseteq \mathbb{R}^n$ is defined to be the set of those points for which associated initial ideals also contains no monomials. An ideal $I$ is said to be well-poised if all of the initial ideals obtained from points in the tropical variety $\mathrm{Trop}(I)$ are prime. It is of interest to know when the ideal is a well-poised ideal. In paper [1] all well-poised principal ideals over an algebraically closed field are classified.

References:
[1] Well-poised hypersurfaces, J. Cecil, N. Dutta, C. Manon, B. Riley, A. Vichitbandha. [arXiv: 2008.00060]