The majorization characterization conjecture for ordinal spaces embeddable in the line

Let $(X,\delta)$ be an ordinal space with $|X|=n\in\mathbb N^+$, and let $x_1,x_2,\ldots,x_n$ be an enumeration of its points satisfying the majorization property. Write $X\hookrightarrow\mathbb R^1$ for embeddability of $X$ in the real line. Majorization characterization conjecture. If such an enumeration exists, then

$$X\hookrightarrow\mathbb R^1.$$

The paper proves that line embeddability implies the existence of an enumeration with the majorization property and states sufficiency for $n\leqslant4$, while explicitly saying that it cannot determine whether sufficiency holds for $n>5$; the general assertion is therefore open.