The regulator-growth conjecture for real quadratic fields

Let $K$ range over real quadratic fields, with regulator $R_K$, discriminant $d_K$, and parameter $m_K$ as used in the source. Regulator-growth conjecture. There exists an infinite sequence of real quadratic fields such that

$$\lim_{m_K\to+\infty}\frac{\log(R_K)}{\log(\sqrt{|d_K|})}=1.$$

This conjecture concerns the possibility that the regulator of real quadratic fields has the same asymptotic logarithmic growth as $\sqrt{|d_K|}$. The supplied context does not define $m_K$ or indicate whether the conjecture is known, so its status remains open.