Half-order immersion conjecture for graphs with independence number at most two

Let $G$ be a graph on $n$ vertices, and let $\alpha(G)$ denote its independence number. An immersion of a graph $H$ in $G$ is a subgraph obtainable from $G$ by lifting edges and deleting vertices and/or edges. Half-order immersion conjecture. Any graph $G$ with $\alpha(G) \leq 2$ contains an immersion of $K_{\lceil n/2 \rceil}$. Since every color class has at most two vertices in such a graph, this conjecture is the natural half-order consequence of Abu-Khzam–Langston's restricted conjecture. The source proves that this statement is equivalent, for this graph class, to the restricted Abu-Khzam–Langston conjecture, but leaves both statements unresolved in general.