DFI equidistribution conjecture for polynomial roots modulo primes

Let $f \in \mathbb{Z}[X]$ be an irreducible polynomial of degree $d \geq 2$, and let $0 < \alpha < \beta \leq 1$. Let $P$ be the set of primes and define

$$S(x):= \{(p,\nu): p \in P,\ p \leq x,\ \nu \in \mathbb{Z},\ 0\leq \nu < p,\ f(\nu) \equiv 0 \pmod p\}.$$

View $S(x)$ as a probability space with normalized counting measure.

DFI conjecture. As $|S(x)| \to \infty$, the probability that $\alpha \leq \frac{\nu}{p} < \beta$ approaches $\beta-\alpha$.

The quadratic case was proved in the work cited by the authors, whereas the general case was described as far away from resolution and remains open.