Jang–Kim's maximal norm conjecture for indecomposable integers

Let $D$ be a positive nonsquare integer, let $\mathbb Z[\sqrt D]$ be the quadratic order, and let $N$ be the minimum of the absolute values of negative norms of elements of $\mathbb Z[\sqrt D]$. Let $a$ be the smallest nonnegative rational integer such that $N$ divides $D-a^2$. An element of $\mathbb Z[\sqrt D]$ is indecomposable if it cannot be written as a sum of two totally positive elements. Jang–Kim's conjecture. For every indecomposable $\alpha\in\mathbb Z[\sqrt D]$, one has

$$N(\alpha)\leq \frac{D-a^2}{N}.$$

The conjecture proposed an improvement of the general bound $N(\alpha)\leq D/N$ for indecomposable integers. It is false in the quadratic field considered in the paper, so the proposed bound does not hold in general.