Minkowski improvement conjecture for ideal classes in totally real fields

Fix an integer $d \geq 3$. For a totally real number field $K$ of degree $d$, let $\operatorname{disc}(K)$ denote its discriminant, and let an ideal class mean a class of fractional ideals of the maximal order $\mathscr{O}_K$. Minkowski improvement conjecture. Every ideal class in a totally real number field of degree $d$ has a representative ideal $J \subset \mathscr{O}_K$ whose norm satisfies

$$N(J)=o\!\left(\sqrt{\operatorname{disc}(K)}\right).$$

This would sharpen Minkowski's bound $N(J)=O(\sqrt{\operatorname{disc}(K)})$ uniformly for fixed degree; the source presents it as conjectural and gives no resolution.