A key lemma to Kirszbraun's theorem for $\mathbb{R}^2$ states the following:

Given any two finite collections of points $x_1,\dots,x_n$ and $x_1',\dots,x_n'$ in $\mathbb{R}^2$ such that $|x_i'x_j'|\le |x_ix_j|$ for all $i,j=1,\dots,n$ and any $x\in \mathbb{R}^2$, it's always possible to find $x'$ in the convex hull of $x_1',\dots,x_n'$ such that $|x'x_k'|\le |xx_k|$ for all $k=1,\dots,n$.

Intuitively, it seems to me that this result could be generalized in the following way:

Given any two finite collections $x_1,\dots,x_n$ and $x_1',\dots,x_n'$ with $1<l<m<n$ such that

(1) $|x_i'x_j'|\le |x_ix_j|$ for all $i,j=1,\dots,m$ and

(2) $|x_k'x_h'|\le |x_kx_h|$ for all $k,\ h=1,\dots,l,\ m+1,\dots,n$,

and any point $x\in Conv(x_1,\dots,x_l)\setminus \{x_1,\dots,x_l\}$, it is always possible to find $x'$ in the convex hull of $x_1',\dots,x_l'$ such that $|x'x_t'|\le|xx_t|$ for all $t=1,\dots,n$.

As it's clear, my idea is to take more than one collection of points which satisfy the hypothesis of Kirszbarun's lemma and which have non empty intersection.

I'm not able to prove it, but it seems right to me, since I can not find any counterexample. So I'd like to ask you if what I'm saying could have sense and, if you think so, if you have an idea of a possible strategy for the proof.

Thank you.