The edge-deletion conjecture for two-eigenvalue realizations of complete graphs

Let $K_n$ be the complete graph on $n$ vertices, and let $H$ be a subgraph of $K_n$ with $|E(H)|\leq n-3$. Write $K_n\backslash H$ for the graph obtained by removing the edges of $H$ from $K_n$. Edge-deletion conjecture. Removing up to $n-3$ edges from $K_n$ does not change its number of distinct eigenvalues; explicitly, for every such $H$,

$$q(K_n\backslash H)=2.$$

This would extend the equality known for $n\leq 7$ and give the conjectured threshold for guaranteeing a two-eigenvalue matrix realization after deleting edges from a complete graph.