Orbit-profile invariance conjecture for Nichols algebras

Let $(X,r)$ be a finite non-degenerate indecomposable solution of the Yang–Baxter equation, and let $W_{X,r}$ be the associated braided vector space. Define the orbit profile

$$\Phi_{X,r}=(l_1,l_2,l_3,\ldots),$$

where $l_n$ is the number of orbits of the action generated by $r$ on $X\times X$ having $n$ elements. Suppose

$$\dim \mathfrak{B}(W_{X,r})=m$$

(or $\operatorname{GKdim}\mathfrak{B}(W_{X,r})=m$). If $(Y,r')$ is any finite non-degenerate solution of the Yang–Baxter equation with

$$\Phi_{X,r}=\Phi_{Y,r'},$$

then $\dim \mathfrak{B}(W_{Y,r'})=m$ (or $\operatorname{GKdim}\mathfrak{B}(W_{Y,r'})=m$) under some given conditions.

Orbit-profile invariance conjecture. Equality of the orbit profiles should preserve the stated dimension, or Gelfand–Kirillov dimension, under the conditions intended in the source.

The proposed principle would make orbit data a useful invariant in studying Nichols algebras associated with non-degenerate solutions of the Yang–Baxter equation. The supplied statement leaves the “given conditions” unspecified, and the source provides no resolution.