Ashrafi's conjecture on the chromatic number of full exponent group power graphs

Let $G$ be a finite group. Its exponent is the least common multiple of the orders of its elements; $G$ is a full exponent group if it contains an element whose order equals the exponent. Suppose that the exponent of $G$ is

$$p_1^{\beta_1}p_2^{\beta_2}\cdots p_r^{\beta_r},$$

where $p_1<p_2<\cdots<p_r$, and let $\omega(G)$ and $\chi(G)$ denote the clique and chromatic numbers of the power graph of $G$. Ashrafi's conjecture. The formula

$$\omega(G)=\chi(G)=p_r^{\beta_r}+\sum_{j=0}^{r-2}\left(p_{r-j-1}^{\beta_{r-j-1}}-1\right)\prod_{i=0}^{j}\phi\left(p_{r-i}^{\beta_{r-i}}\right)$$

is correct in general.

This conjecture extends the stated theorem for full exponent groups from the cited work of Ashrafi and coauthors. The paper's abstract says that the corresponding Beck conjecture for power graphs is answered affirmatively in complete generality, but the supplied excerpt gives no explicit resolution sentence for this particular conjecture; its database status is therefore left open pending verification against the paper's proof.