Quadratic EPPA lower bound outside finite homogeneous subgraphs

Let $G$ be a graph on $n$ vertices. A finite graph is homogeneous if every isomorphism between finite induced subgraphs extends to an automorphism, and an EPPA-witness of $G$ is a finite graph extending all partial automorphisms of $G$.

Quadratic EPPA conjecture. The graph $G$ is either a subgraph of a finite homogeneous graph, or every EPPA-witness of $G$ has at least

$$\Omega(n^2)$$

vertices.

This would improve the general lower bound discussed in the paper from a weaker estimate to a quadratic one. The source gives no resolution.