Given a positive integer $n\in \mathbb{N}$ we define the Hadwiger-Nelson graph $\text{HN}_n$ by

1. $V(\text{HN}_n) = \mathbb{R}^n$;
2. $E(\text{HN}_n) = \{\{v_1, v_2\}: v_1, v_2 \in \mathbb{R}^n \text{ and } |v_1-v_2| = 1\}$.

Is it true that the clique number $\omega(\text{HN}_n)$ equals $n+1$ for all $n\in \mathbb{N}$?