O. Nguyen's intermediate real-root count conjecture for random polynomials

Let $(\varepsilon_k)$ be an i.i.d. sequence of standard Gaussian random variables, and let

$$f_n(x)=\sum_{k=0}^n c_k\varepsilon_k x^k$$

with $(c_k)$ a bounded sequence of real numbers. For $\alpha\in(0,1)$, write $R_n$ for the number of real roots of $f_n$.

O. Nguyen's conjecture. Given $\alpha\in(0,1)$, there exists a bounded sequence $(c_k)$ of real numbers such that

$$\mathbb E R_n=n^{\alpha+o(1)}.$$

This conjecture asks whether random polynomials with Gaussian coefficients can realize every intermediate power-law order of expected real roots between the logarithmic and linear regimes. The surrounding discussion cites known logarithmic, square-root, and linear-root regimes, but provides no resolution of the conjecture.