Graph-theoretic partition conjecture for quantum-code connectivity graphs

Let $\beta \ge 1$ and $c\in (0,1)$ be constants. There exists a constant $\alpha=\alpha(\beta,c)$ such that the following holds for all $n\ge d$. Let $G$ be a graph with separation profile $s_G(r)\le \beta r^c$ for all $r$. A set of vertices is $d$-correctable when it is correctable with respect to graph $G$ in the sense used for the quantum code.

Graph theory conjecture. There exists a partition

$$A\sqcup B\sqcup C$$

of $G$ such that $A$ and $B$ are $d$-correctable with respect to graph $G$ and

$$|C|\le \alpha n/d^{2(1-c)/c}.$$

This conjecture would provide the stronger partition needed to derive the improved rate-distance trade-off. The source gives no proof or resolution.