Exact depth-three bound for two-colored random tournaments

Let $f^*(2,3,n)$ denote the random variable measuring the largest number of vertices that can be reached from a vertex by monochromatic directed paths of depth at most $3$ in a two-edge coloring of a random $n$-vertex tournament. Depth-three exactness conjecture. With high probability,

$$f^*(2,3,n)=n.$$

This is the quantitative form of the preceding depth-three monochromatic domination conjecture: equality means that some vertex reaches every other vertex. It remains open in the source.