Yan et al.'s Ohba-like conjecture for improper colorings

Let $G$ be a graph and let $d$ be a nonnegative integer. Let $\chi^d(G)$ and $\chi_l^d(G)$ denote respectively the $d$-improper chromatic number and the $d$-improper list chromatic number of $G$. Yan et al.'s conjecture. If

$$|V(G)|\le (d+2)\chi^d(G)+(d+1),$$

then

$$\chi_l^d(G)=\chi^d(G).$$

The source attributes this conjecture to Yan et al.; no resolution evidence is supplied here. The paper notes that its generalized Ohba conjecture would imply this statement.