Vergara's conjecture for graphs with independence number two

Let $G$ be a graph with independence number $\alpha(G)=2$, let $\chi(G)$ denote its chromatic number, and let $K_{\chi(G)}$ be the complete graph on $\chi(G)$ vertices. A weak immersion of a graph $H$ in $G$ consists of an injection from $V(H)$ to $V(G)$ and pairwise edge-disjoint paths in $G$ representing the edges of $H$. Vergara's conjecture. Every graph $G$ with $\alpha(G)=2$ contains a weak immersion of $K_{\chi(G)}$. The source presents this as the Abu-Khzam–Langston conjecture restricted to graphs of independence number two and notes its equivalence, for graphs with $\alpha(G)\leq 2$, to the corresponding assertion involving $K_{\lceil |V(G)|/2\rceil}$. The status of this named formulation is not separately established in the supplied text.