Conjecture on large co-edge-regular graphs with four eigenvalues

Let $\Gamma$ be a connected co-edge-regular graph with four distinct eigenvalues. Let $t\geq 2$ be an integer and write $|V(\Gamma)|=n(\Gamma)$. An $s$-clique extension is a graph obtained from a graph by replacing each vertex with a clique of size $s$, with adjacency between cliques determined by adjacency of the corresponding original vertices. Large co-edge-regular graph conjecture. There exists a constant $n_t$ such that, if

$$\theta_{\min}(\Gamma)\geq -t \quad\text{and}\quad n(\Gamma)\geq n_t,$$

then $\Gamma$ is the $s$-clique extension of a strongly regular graph for some integer $s$ satisfying $2\leq s\leq t-1$. This predicts a structural classification of sufficiently large connected co-edge-regular graphs in this spectral range; the source presents it as a belief motivated by the study of regular graphs with four distinct eigenvalues and gives no resolution.