The smooth-value asymptotic for polynomial values

Let $F(x)\in\mathbb{Z}[x]$, and let $\Psi_F^*(x,y)$ denote the counting function for the relevant integers whose values under $F$ satisfy the smoothness condition encoded by the source. Smooth-value conjecture. The asymptotic

$$\Psi_F^*(x,x^{\alpha})\asymp_{F,\alpha}\frac{x}{\log x}$$

holds for every positive $\alpha$. This is quoted from DMT as an expected strengthening of the known estimate; the supplied text gives no resolution status.