The Brauer–Siegel conjecture for families of number fields

Let $K_i$ be number fields, with degree $n_{K_i}=[K_i:\mathbb{Q}]$, absolute discriminant $d_{K_i}=|\operatorname{disc}(K_i/\mathbb{Q})|$, class number $h_{K_i}$, regulator $R_{K_i}$, and $g_{K_i}=\log\sqrt{d_{K_i}}$. Let $\rho_{K_i}$ denote the residue of $\zeta_{K_i}(s)$ at $s=1$. Assume

$$d_{K_i}^{1/n_{K_i}}\longrightarrow\infty.$$

Brauer–Siegel conjecture. Then

$$\lim_{i\to\infty}\frac{\log h_{K_i}R_{K_i}}{g_{K_i}}=1.$$

Equivalently,

$$\lim_{i\to\infty}\frac{\log\rho_{K_i}}{g_{K_i}}=0.$$

The conjecture is expected in general; the paper notes that existing proofs cover several structured families, including Galois, almost normal, and solvable-Galois-closure families, while the general case remains open.