The colinearity and singular-set conjecture for homogeneous chambars

Let $R$ be the radial vector field on $\mathbb{C}^n$, and let

$$\mathrm{Col}(X,Y)=\{m\in\mathbb{C}^n\mid X(m)\wedge Y(m)=0\}$$

be the set of colinearity of two holomorphic vector fields. Let $\mathrm{Ch}(X_1,X_2,\ldots,X_p)$ be a homogeneous $p$-chambar of degree $\nu\geq 1$ on $\mathbb{C}^n$, with $p\geq 2$; write $\mathrm{Sing}(X_k)=\{m\in\mathbb{C}^n\mid X_k(m)=0\}$.

Homogeneous-chambar singular-set conjecture. For every $k\geq 1$,

$$\mathrm{Col}(X_k,R)=\mathrm{Sing}(X_k).$$

In particular,

$$\dim\mathrm{Sing}(X_k)\geq 1.$$

The conjecture concerns the geometry of homogeneous chambars and predicts that each vector field is colinear with the radial field exactly on its singular set. The source gives no resolution.