Constant Values on Arrangements of Points under Maps $S^n\to\mathbb{R}^m$

Question

The following result would be a generalisation of this result of Floyd, as well as The Borsuk-Ulam Theorem. Let $X\subseteq S^n$. Call $[X]$ the arrangement given by $X$, defined as the equivalence class of all sets $Y\subseteq S^n$ such that there is a rotation $\rho$ of $S^n$ such that $\rho(X)=Y$. Is it the case that for every continuous $f:S^n\to\mathbb{R}^m$, and every $(n-m+2)$-subset $X$ of $S^n$ there exists $Y\in[X]$ and a constant $c\in\mathbb{R}^m$ such that for all $s\in Y$, $f(s)=c$? This is at least true for $m=1$ and $n=1,2$.