Polynomial concentration conjecture for the number of real zeros of Kac polynomials

Let $f_n(x)=\sum_{i=0}^{n}\xi_i x^i$ be a Kac polynomial whose independent coefficients $\xi_i$ have mean zero and variance one, and let $N_n(\mathbb R)$ denote its number of real roots. Under mild conditions on the coefficients, for every $\varepsilon>0$ there exist constants $c,C>0$ such that, for all $n$,

$$\frac{1}{C}n^{-C}\leq \mathbb P\left(\left|N_n(\mathbb R)-\mathbb E N_n(\mathbb R)\right|\geq \varepsilon\log n\right)\leq Cn^{-c}.$$

Concentration conjecture. Under mild conditions, the probability that the number of real zeros deviates from its expectation by at least $\varepsilon\log n$ is bounded above and below by polynomial functions of $n$ as displayed above. This would provide quantitative concentration estimates complementing the known logarithmic asymptotics for the expected number of real roots and the central limit theorem for $N_n(\mathbb R)$.