Here is a proof to a related claim that hopefully will give you some ideas. 

**Claim.** 
Let $X$ be an equivalence class of the entanglement relation on $V(G)$, with
$|X| \geq 2$.  Then for all distinct $u,v \in X$, there exist
$\chi(G)-1$ edge-disjoint paths in $G$ between $u$ and $v$.

*Proof.* Let $k=\chi(G)$ and $V_1, \dots, V_k$ be a partition of $V(G)$ into stable sets.  By relabelling, we may assume that $X \subseteq V_1$. Observe that all vertices of $X$ must be contained in some component of $G[V_1 \cup V_2]$. Otherwise, we may recolour to obtain a $k$-colouring of $G$ where two vertices of $X$ are coloured differently.  In particular, for all distinct $u,v \in X$, there is a $u$--$v$ path in $G[V_1, \cup V_2]$.  Repeating the argument for $i=2, \dots, k$, gives the $k-1$ edge-disjoint $u$--$v$ paths in $G$. $\square$

Note that the claim proves something stronger and weaker than what was asked in the original question.  It is weaker because the paths are *edge-disjoint* not vertex-disjoint.  But it is stronger since it holds for *all* distinct pairs $u,v \in X$.  Moreover, the paths constructed in the proof are almost vertex-disjoint.  The only vertices they have in common are in $V_1$.