Power-sum character formula conjecture for cyclic Eulerian representations

Let $\lambda=(\lambda_1,\ldots,\lambda_k)\vdash n$, let $g(\lambda)=\gcd(\lambda_1,\ldots,\lambda_k)$, and let $A_{k-1}(t)$ and $[\lambda_i]_t$ have the meanings defined in the paper. For a polynomial $F(t)$ and positive integer $m$, let $F(t)_m$ be obtained by retaining only terms $a_it^i$ with $\gcd(m,i)=1$. Define

$$G_\lambda(t)=\left(tA_{k-1}(t)\prod_{i=1}^k[\lambda_i]_t\right)_{g(\lambda)}.$$

Power-sum character formula conjecture. The Eulerian quasisymmetric functions satisfy

$$\sum_{j=0}^{n-1}Q_{(n),j}t^j=\sum_{\lambda\vdash n}z_\lambda^{-1}G_\lambda(t)p_\lambda.$$

Equivalently, the character of $V_{(n),j}$ evaluated on the conjugacy class $\lambda$ is the coefficient of $t^j$ in $G_\lambda(t)$. The source says this conjecture was proved by Sagan, Shareshian and Wachs and had been verified there through $n=8$.