Is the following fact true?

Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k \in \mathbb{R}^2$, $\|w_i\|\leq 1$, such that $v_{\sigma(i)}=w_i-w_{i-1}$? (here, I assume $w_0=w_k$)

**Edit. Related (known) facts:**

1. The same fact in $\mathbb{R}^1$ is true. (can be easily proven by choosing $w_0=0$ and $\sigma(i)$ such that $\|w_i\|\leq 1$ for $w_i:=w_{i-1}+v_{\sigma(i)}$)

2. For each $\epsilon>0$ there exists a family of vectors $v_i$, such that $\max_i\|w_i\|>1-\epsilon$. See my comment below for the proof.