The finite Menger-type embedding conjecture for ordinal spaces

Let $(X,\delta)$ be a finite ordinal space and let $n\in\mathbb N^+$. For a subset $A\subseteq X$, write $A\hookrightarrow\mathbb R^n$ when the induced ordinal space on $A$ embeds in $\mathbb R^n$. Finite Menger-type embedding conjecture. The following conditions are equivalent:

• $X\hookrightarrow\mathbb R^n$;
• for every $A\subseteq X$ with $|A|\leqslant n+3$, one has $A\hookrightarrow\mathbb R^n$.

This is proposed as the ordinal-space analogue of Menger's finite-subset criterion for abstract semimetric spaces. The paper presents it as a question-inspired conjecture and supplies no proof or counterexample in the given text.