The proposed classification of 4-chambars on an open subset of the complex line

A $4$-chambar is a chambar consisting of four vector fields on an open subset of $\mathbb{C}$; two such objects are affinely conjugate when related by an affine change of coordinate. A chambar is rigid if it has no nontrivial deformations, and special denotes the class described in the statement. The $t$-degree refers to the degree in $t$ of the associated flow.

4-chambar classification conjecture. Up to affine conjugacy, every $4$-chambar on an open subset of $\mathbb{C}$ is one of the following types:

• constant $\mathrm{Ch}\left(a_1\frac{\partial}{\partial x},a_2\frac{\partial}{\partial x},a_3\frac{\partial}{\partial x},a_4\frac{\partial}{\partial x}\right)$, with $a_k\in\mathbb{C}^*$;
• rigid of $t$-degree $2$: $\mathrm{Ch}(a_1X,a_2X,a_3X,a_4X)$, where $X=2\sqrt{x}\,\frac{\partial}{\partial x}$ and

$$a_1+a_2+a_3+a_4=a_1^2+a_2^2+a_3^2+a_4^2=0;$$

• rigid of $t$-degree $3$: $\mathrm{Ch}(X,\sigma X,\sigma^2X,\sigma^3X)$, where $X$ has $t$-degree $3$ and $\sigma$ is a fourth root of unity;
• special $\mathrm{Ch}(X,-X,Y,-Y)$, where $X$ and $Y$ have $t$-degree $2$.

The claim is a classification of the four-element configurations considered in the paper. The supplied source gives no resolution, although the surrounding discussion presents the list as a proposed classification.