I have just found out a very similar theorem to the one in the question. Some work still must be done, but it seems clear that it is very closely related.

https://arxiv.org/pdf/1806.08303.pdf 

Let $G = (V, E)$ be a simple graph. For $B$ a subset of the vertex set $V$ , we define the spread of $B$ as 
$$\mathrm{sp}(B) = \Big\{\max[\mathrm{deg}(u)]−
\min[\mathrm{deg}(v)] : u, v ∈ B\Big\}.$$
We then define, for an integer $k ≥ 0$, quantity $\mathrm{sp}(G, k)$ as:
$$\max{\Big\{|B|: \ {sp(B) ≤ k}\Big\}},$$
i.e. "the largest cardinality of a subset of vertices of $G$ with spread at most $k$".

Now, we have the following,

**Theorem** (*Erdős, Chen, Rousseau and Schelp*):

> Let $G$ be a graph on $n ≥ k+1$ vertices, then $\mathrm{sp}(G, k-1) ≥ k + 1.$