Non-universality conjecture for the first non-universal term in Kac polynomial real zeros

Let $P_n(x)=\sum_{k=0}^{n}\xi_kx^k$ be a Kac random polynomial, where the coefficients $\xi_k$ are independent and identically distributed copies of a random variable $\xi$. Assume that $\xi$ has mean zero, variance one, and a finite $(2+\varepsilon_0)$-moment for some $\varepsilon_0>0$. Let $N_{\mathbb{R}}(P_n)$ denote the number of real roots of $P_n$. Non-universality conjecture. There exists a constant $C_{\xi}$ depending only on $\xi$ such that

$$\mathbb{E}[N_{\mathbb{R}}(P_n)]=\frac{2}{\pi}\log n+C_{\xi}+o(1)\qquad\text{as }n\to\infty.$$

Moreover, $C_{\xi}$ is non-universal; in particular, it may differ from the Gaussian constant $C_{\mathrm{Gau}}$ when $\xi$ is non-Gaussian. The leading term is known to be universal, with the expected number of real roots equal to $\frac{2}{\pi}\log n+O(1)$ under these moment assumptions, while the conjectured constant term is expected to retain information about the coefficient distribution. Its existence and non-universality beyond the Gaussian case remain open.