The skewed coloring sampling conjecture

Let $G$ be a graph on $n$ vertices with maximum degree $\Delta$, and let $\vec{n}=(n_1,\ldots,n_q)$ satisfy $\|\vec{n}\|_1=n$. An $\vec{n}$-coloring is a proper coloring with exactly $n_i$ vertices of color $i$.

Skewed sampling conjecture. There should be a polynomial-time algorithm that approximately samples uniformly from the set of $\vec{n}$-colorings of $G$ whenever

$$n_i\leq\left\lfloor\dfrac{n}{2\Delta}\right\rfloor$$

for every $i\in[q]$.

The paper presents this as a possible first step toward sampling substantially more skewed colorings, requiring new ideas; no resolution is supplied.