Tan et al.'s conjecture on co-edge-regular graphs with four eigenvalues

Let $G$ be a connected $k$-regular graph with $n$ vertices and co-edge-regular with parameter $\mu$, having four distinct eigenvalues. Let $t\geq 2$ be an integer. Tan et al.'s conjecture. There exists a constant $n_t$ such that, if $\theta_{\min}(G)\geq -t$, $n\geq n_t$ and $k<n-2-\frac{(t-1)^2}{4}$, then either $G$ is the $s$-clique extension of a strongly regular graph for $2\leq s\leq t-1$, or $G$ is a $p\times q$-grid with $p>q\geq 2$. The conjecture seeks a Neumaier-type classification for co-edge-regular graphs with four distinct eigenvalues; the cited spectral characterizations of clique extensions motivate it, but no resolution is supplied here.