The conjectured threshold for graphs with two distinct eigenvalues

For each integer $n\geq 2$, let $m(n)$ denote the least number of edges such that every graph on $n$ vertices with at least $m(n)$ edges has a matrix realization with two distinct eigenvalues. Threshold conjecture.

$$m(2)=1,$$

and

$$m(n)={n\choose 2}-(n-3)$$

for all $n\geq 3$. The values through $n=7$ support this formula, while the preceding lower bound establishes only that $m(n)\geq {n\choose 2}-(n-3)$ in the stated range.