Existence conjecture for polynomial flows with prescribed heteroclinic regions

Consider the holomorphic flow

$$\dot{x}=F(x),$$

where $F$ is a complex polynomial, an equilibrium is a zero of $F$, and an equilibrium has order $m$ when it is a zero of multiplicity $m$. A heteroclinic region is a strip region whose boundary contains orbits connecting distinct equilibria. For all $n,m\in\mathbb{N}$, there should exist a complex polynomial $F_{n,m}$ of degree $n+m$ such that the flow has an equilibrium $a$ of order $m$ and $n$ foci or nodes $a_1,\ldots,a_n$ satisfying the following properties: either all equilibria $a_1,\ldots,a_n$ are stable or all are unstable; for every $j\in\{1,\ldots,n\}$, there is a heteroclinic region between $a$ and $a_j$; and all heteroclinic orbits tend to $a$ in the same definite direction.

Existence conjecture for polynomial flows with prescribed heteroclinic regions. For all $n,m\in\mathbb{N}$, there exists a complex polynomial $F_{n,m}$ of degree $n+m$ such that the associated flow has an equilibrium $a$ of order $m$ and $n$ foci or nodes $a_1,\ldots,a_n$ with properties (i)--(iii) above.

The conjecture asks for examples realizing arbitrarily many heteroclinic regions between one multiple equilibrium and foci or nodes while controlling their stability and the common direction in which heteroclinic orbits approach the multiple equilibrium. The source states that a general example is still lacking, so the conjecture remains open.