Hooley-measure equidistribution conjecture for polynomial roots modulo primes

Let $f\in\mathbf{Z}[X]$ be a monic irreducible polynomial of degree at least $2$. For each prime $p$, let $\varrho_f(p)$ be the number of roots of $f$ modulo $p$, and let $\Delta_p$ be the uniform probability measure on those roots. Hooley-measure equidistribution conjecture. The measures

$$\frac{1}{\pi(x)}\sum_{\substack{p\leqslant x\p\in\mathcal{Q}}}\varrho_f(p)\Delta_p$$

converge to the uniform measure as $x\to+\infty$. This is the natural conjecture using Hooley's measures; the source notes that it was stated elsewhere and gives no resolution here.